JEE Main & Advanced Mathematics Functions Question Bank

done Functions Total Questions - 165

• question_answer1)
  sIf \[f(x)=\cos (\log x)\], then \[f(x)f(y)-\frac{1}{2}[f(x/y)+f(xy)]=\] [IIT 1983; RPET 1995; MP PET 1995; Karnataka CET 1999; UPSEAT 2001]
  A)
             \[-1\] done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             \[-2\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer2)
  If \[f(x)=\frac{1-x}{1+x},\] the  n \[f[f(\cos \ 2\theta )]=\] [MP PET 1994, 2001; Pb. CET 2002]
  A)
             \[\tan 2\theta \] done clear
  B)
             \[\sec 2\theta \] done clear
  C)
             \[\cos 2\theta \] done clear
  D)
             \[\cot 2\theta \] done clear
  View Solution play_arrow

• question_answer3)
  If \[f(x)=\sin \log x\], then the value of \[f(xy)+f\left( \frac{x}{y} \right)-2f(x).\cos \log y\] is equal to     [Orissa JEE 2004]
  A)
             1 done clear
  B)
             0 done clear
  C)
             ?1 done clear
  D)
             \[\sin \log x.\cos \log y\] done clear
  View Solution play_arrow

• question_answer4)
  The value of b and c for which the identity \[f(x+1)-f(x)=8x+3\] is satisfied, where \[f(x)=b{{x}^{2}}+cx+d\],  are                                                                         [Roorkee 1992]
  A)
             \[b=2,\ c=1\] done clear
  B)
             \[b=4,\ c=-1\] done clear
  C)
             \[b=-1,\ c=4\] done clear
  D)
             \[b=-1,\ c=1\] done clear
  View Solution play_arrow

• question_answer5)
  Given the function \[f(x)=\frac{{{a}^{x}}+{{a}^{-x}}}{2},\ (a>2)\]. Then \[f(x+y)+f(x-y)=\]
  A)
             \[2f(x).f(y)\] done clear
  B)
             \[f(x).f(y)\] done clear
  C)
             \[\frac{f(x)}{f(y)}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer6)
  If \[f(x)=\frac{x}{x-1}\], then \[\frac{f(a)}{f(a+1)}=\]                       [MP PET 1996]
  A)
             \[f(-a)\] done clear
  B)
             \[f\left( \frac{1}{a} \right)\] done clear
  C)
             \[f({{a}^{2}})\] done clear
  D)
             \[f\left( \frac{-a}{a-1} \right)\] done clear
  View Solution play_arrow

• question_answer7)
  If \[f(x)=\cos (\log x)\], then \[f({{x}^{2}})f({{y}^{2}})-\frac{1}{2}\left[ f\,\left( \frac{{{x}^{2}}}{2} \right)+f\left( \frac{{{x}^{2}}}{{{y}^{2}}} \right) \right]\] has the value                                 [MNR 1992]
  A)
             ?2 done clear
  B)
             ?1 done clear
  C)
             ½ done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer8)
  The equivalent function of \[\log {{x}^{2}}\] is [MP PET 1997]
  A)
             \[2\log x\] done clear
  B)
             \[2\log |x|\] done clear
  C)
             \[|\log {{x}^{2}}|\] done clear
  D)
             \[{{(\log x)}^{2}}\] done clear
  View Solution play_arrow

• question_answer9)
  If \[f(x)=\log \left[ \frac{1+x}{1-x} \right]\], then \[f\left[ \frac{2x}{1+{{x}^{2}}} \right]\] is equal to [MP PET 1999; RPET 1999; UPSEAT 2003]
  A)
             \[{{[f(x)]}^{2}}\] done clear
  B)
             \[{{[f(x)]}^{3}}\] done clear
  C)
             \[2f(x)\] done clear
  D)
             \[3f(x)\] done clear
  View Solution play_arrow

• question_answer10)
  If \[\varphi (x)={{a}^{x}}\], then \[{{\{\varphi (p)\}}^{3}}\]is  equal to [MP PET 1999]
  A)
             \[\varphi (3p)\] done clear
  B)
             \[3\varphi (p)\] done clear
  C)
             \[6\varphi (p)\] done clear
  D)
             \[2\varphi (p)\] done clear
  View Solution play_arrow

• question_answer11)
  If \[f(x)=\frac{x-3}{x+1}\], then \[f[f\{f(x)\}]\] equals [RPET 1996]
  A)
             x done clear
  B)
             ?x done clear
  C)
             \[\frac{x}{2}\] done clear
  D)
             \[-\frac{1}{x}\] done clear
  View Solution play_arrow

• question_answer12)
  If \[f(x)=\cos (\log x)\], then the value of \[f(x).f(4)-\frac{1}{2}\left[ f\left( \frac{x}{4} \right)+f(4x) \right]\] [Kurukshetra CEE 1998]
  A)
             1 done clear
  B)
             ?1 done clear
  C)
             0 done clear
  D)
             \[\pm 1\] done clear
  View Solution play_arrow

• question_answer13)
  If \[f(x)=\frac{x-|x|}{|x|}\], then \[f(-1)=\] [SCRA 1996]
  A)
             1 done clear
  B)
             ?2 done clear
  C)
             0 done clear
  D)
             +2 done clear
  View Solution play_arrow

• question_answer14)
  If \[f(x)=4{{x}^{3}}+3{{x}^{2}}+3x+4\], then \[{{x}^{3}}f\left( \frac{1}{x} \right)\] is            [SCRA 1996]
  A)
             \[f(-x)\] done clear
  B)
             \[\frac{1}{f(x)}\] done clear
  C)
             \[{{\left( f\left( \frac{1}{x} \right) \right)}^{2}}\] done clear
  D)
             \[f(x)\] done clear
  View Solution play_arrow

• question_answer15)
  Let \[f:R\to R\] be defined by \[f(x)=2x+|x|\], then \[f(2x)+f(-x)-f(x)=\] [EAMCET 2000]
  A)
             \[2x\] done clear
  B)
             \[2|x|\] done clear
  C)
             \[-2x\] done clear
  D)
             \[-2|x|\] done clear
  View Solution play_arrow

• question_answer16)
  If \[f(x+ay,\ x-ay)=axy\], then \[f(x,\ y)\] is equal to            [AMU 2001]
  A)
             xy done clear
  B)
             \[{{x}^{2}}-{{a}^{2}}{{y}^{2}}\] done clear
  C)
             \[\frac{{{x}^{2}}-{{y}^{2}}}{4}\] done clear
  D)
             \[\frac{{{x}^{2}}-{{y}^{2}}}{{{a}^{2}}}\] done clear
  View Solution play_arrow

• question_answer17)
  If \[f(x)=\cos [{{\pi }^{2}}]x+\cos [-{{\pi }^{2}}]x\], then [Orissa JEE 2002]
  A)
             \[f\left( \frac{\pi }{4} \right)=2\] done clear
  B)
             \[f(-\pi )=2\] done clear
  C)
             \[f(\pi )=1\] done clear
  D)
             \[f\left( \frac{\pi }{2} \right)=-1\] done clear
  View Solution play_arrow

• question_answer18)
  If \[f(x)=\frac{1}{\sqrt{x+2\sqrt{2x-4}}}+\frac{1}{\sqrt{x-2\sqrt{2x-4}}}\] for \[x>2\], then \[f(11)=\] [EAMCET 2003]
  A)
             7/6 done clear
  B)
             5/6 done clear
  C)
             6/7 done clear
  D)
             5/7 done clear
  View Solution play_arrow

• question_answer19)
  If \[{{e}^{f(x)}}=\frac{10+x}{10-x},\ x\in (-10,\ 10)\] and \[f(x)=kf\left( \frac{200x}{100+{{x}^{2}}} \right)\], then \[k=\] [EAMCET 2003]
  A)
             0.5 done clear
  B)
             0.6 done clear
  C)
             0.7 done clear
  D)
             0.8 done clear
  View Solution play_arrow

• question_answer20)
  If \[f(x)=2\sin x\], \[g(x)={{\cos }^{2}}x\], then \[(f+g)\left( \frac{\pi }{3} \right)=\]
  A)
             1 done clear
  B)
             \[\frac{2\sqrt{3}+1}{4}\] done clear
  C)
             \[\sqrt{3}+\frac{1}{4}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer21)
  The graph of the function \[y=f(x)\] is symmetrical about the line \[x=2\], then                                 [AIEEE 2004]
  A)
             \[f(x)=-f(-x)\] done clear
  B)
             \[f(2+x)=f(2-x)\] done clear
  C)
             \[f(x)=f(-x)\] done clear
  D)
             \[f(x+2)=f(x-2)\] done clear
  View Solution play_arrow

• question_answer22)
  If \[f(x)=\frac{x}{x-1}=\frac{1}{y}\], then \[f(y)=\] [MP PET 1995, 97]
  A)
             x done clear
  B)
             \[x+1\] done clear
  C)
             \[x-1\] done clear
  D)
             \[1-x\] done clear
  View Solution play_arrow

• question_answer23)
  If \[y=f(x)=\frac{ax+b}{cx-a}\], then x is equal to [AMU 2001]
  A)
             \[1/f(x)\] done clear
  B)
             \[1/f(y)\] done clear
  C)
             \[yf(x)\] done clear
  D)
             \[f(y)\] done clear
  View Solution play_arrow

• question_answer24)
  If \[f(x)=\frac{{{x}^{2}}-1}{{{x}^{2}}+1}\], for every real numbers. then the minimum value of f [Pb. CET 2001]
  A)
             Does not exist because f is bounded                             done clear
  B)
             Is not attained even through f is bounded done clear
  C)
             Is equal to +1                        done clear
  D)
             Is equal to ?1 done clear
  View Solution play_arrow

• question_answer25)
  \[f(x,\ y)=\frac{1}{x+y}\] is a homogeneous function of degree [Orissa JEE 2004]
  A)
             1 done clear
  B)
             ­?1 done clear
  C)
             2 done clear
  D)
             ?2 done clear
  View Solution play_arrow

• question_answer26)
  Let x be a non-zero rational number and y be an irrational number. Then xy is                    [Orissa JEE 2004]
  A)
             Rational done clear
  B)
             Irrational done clear
  C)
             Non-zero done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer27)
  Numerical value of the expression \[\left| \ \frac{3{{x}^{3}}+1}{2{{x}^{2}}+2}\  \right|\] for \[x=-3\] is [Orissa JEE 2004; UPSEAT 2004]
  A)
             4 done clear
  B)
             2 done clear
  C)
             3 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer28)
  The function \[f:R\to R,\ f(x)={{x}^{2}},\forall x\in R\] is [MP PET 1997]
  A)
             Injection but not surjection done clear
  B)
             Surjection but not injection done clear
  C)
             Injection as well as surjection done clear
  D)
             Neither injection nor surjection done clear
  View Solution play_arrow

• question_answer29)
  If for two functions g and f, gof is both injective and surjective, then which of the following is true  [Kurukshetra CEE 1998]
  A)
             g and f should be injective and surjective done clear
  B)
             g should be injective and surjective done clear
  C)
             f should be injective and surjective done clear
  D)
             None of them may be surjective and injective done clear
  View Solution play_arrow

• question_answer30)
  The function which map [?1, 1] to [0, 2] are [Kurukshetra CEE 1998]
  A)
             One linear function done clear
  B)
             Two linear function done clear
  C)
             Circular function done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer31)
  Let \[f(x)=\left\{ \begin{align}   & \frac{1}{2},\ if\ 0\le x\le \frac{1}{2} \\  & \frac{1}{3},\ if\ \frac{1}{2}<x\le 1 \\ \end{align} \right.\], then f is [SCRA 1996]
  A)
             A rational function done clear
  B)
             A trigonometric function done clear
  C)
             A step function done clear
  D)
             An exponential function done clear
  View Solution play_arrow

• question_answer32)
  Function \[f:R\to R,\ f(x)={{x}^{2}}+x\] is [RPET 1999]
  A)
             One-one onto done clear
  B)
             One-one into done clear
  C)
             Many-one onto done clear
  D)
             Many-one into done clear
  View Solution play_arrow

• question_answer33)
  Mapping \[f:R\to R\] which is defined as \[f(x)=\cos x,\ x\in R\] will be [UPSEAT 1999]
  A)
             Neither one-one nor onto done clear
  B)
             One-one done clear
  C)
             Onto done clear
  D)
             One-one onto done clear
  View Solution play_arrow

• question_answer34)
  The function \[f:R\to R\] defined by \[f(x)=(x-1)\] \[(x-2)(x-3)\]  is [Roorkee 1999]
  A)
                     One-one but not onto done clear
  B)
             Onto but not one-one done clear
  C)
             Both one-one and onto done clear
  D)
             Neither one-one nor onto done clear
  View Solution play_arrow

• question_answer35)
  . If \[f:R\to R\], then \[f(x)=\ |x|\] is [RPET 2000]
  A)
                     One-one but not onto done clear
  B)
             Onto but not one-one done clear
  C)
             One-one and onto done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer36)
  Which of the four statements given below is different from others [UPSEAT 2000]
  A)
                     \[f:A\to B\] done clear
  B)
             \[f:x\to f(x)\] done clear
  C)
             f is a mapping of A into B done clear
  D)
             f is a function of A into B done clear
  View Solution play_arrow

• question_answer37)
  Let \[f:N\to N\] defined by \[f(x)={{x}^{2}}+x+1\], \[x\in N\], then f is [AMU 2000]
  A)
                     One-one onto done clear
  B)
             Many one onto done clear
  C)
             One-one but not onto done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer38)
  Let X and Y be subsets of R, the set of all real numbers. The function \[f:X\to Y\]defined by \[f(x)={{x}^{2}}\] for \[x\in X\] is one-one but not onto if (Here \[{{R}^{+}}\] is the set of all positive real numbers)                 [EAMCET 2000]
  A)
                     \[X=Y={{R}^{+}}\] done clear
  B)
             \[X=R,\ Y={{R}^{+}}\] done clear
  C)
             \[X={{R}^{+}},\ Y=R\] done clear
  D)
             \[X=Y=R\] done clear
  View Solution play_arrow

• question_answer39)
  Set A has 3 elements and set B has 4 elements. The number of injection that can be defined from A to B is  [UPSEAT 2001]
  A)
             144 done clear
  B)
             12 done clear
  C)
             24 done clear
  D)
             64 done clear
  View Solution play_arrow

• question_answer40)
  Let \[f:R\to R\] be a function defined by \[f(x)=\frac{x-m}{x-n}\], where \[m\ne n\]. Then [UPSEAT 2001]
  A)
                     f is one-one onto done clear
  B)
             f is one-one into done clear
  C)
             f is many one onto done clear
  D)
             f is many one into done clear
  View Solution play_arrow

• question_answer41)
  The function \[f:R\to R\] defined by \[f(x)={{e}^{x}}\] is [Karnataka CET 2002; UPSEAT 2002]
  A)
             Onto done clear
  B)
             Many-one done clear
  C)
             One-one and into done clear
  D)
             Many one and onto done clear
  View Solution play_arrow

• question_answer42)
  Which one of the following is a objective function on the set of real numbers                                     [Kerala (Engg.) 2002]
  A)
             \[2x-5\] done clear
  B)
             \[|x|\] done clear
  C)
             \[{{x}^{2}}\] done clear
  D)
             \[{{x}^{2}}+1\] done clear
  View Solution play_arrow

• question_answer43)
  Let \[f(x)=\frac{{{x}^{2}}-4}{{{x}^{2}}+4}\] for \[|x|\ >2\], then the function \[f:(-\infty ,\ -2]\cup [2,\ \infty )\to (-1,\ 1)\] is [Orissa JEE 2002]
  A)
             One-one into done clear
  B)
             One-one onto done clear
  C)
             Many one into done clear
  D)
             Many one onto done clear
  View Solution play_arrow

• question_answer44)
  Let the function \[f:R\to R\] be defined by \[f(x)=2x+\sin x,\ x\in R\]. Then f is [IIT Screening 2002]
  A)
             One-to-one and onto       done clear
  B)
                     One-to-one but not onto done clear
  C)
             Onto but not one-to-one done clear
  D)
             Neither one-to-one nor onto done clear
  View Solution play_arrow

• question_answer45)
  A function f from the set of natural numbers to integers defined by \[f(n)=\left\{ \begin{align}   & \frac{n-1}{2},\ \text{when}\ n\ \text{is}\ \text{odd} \\  & -\frac{n}{2},\ \text{when }n\text{ is even} \\ \end{align} \right.\], is [AIEEE 2003]
  A)
                     One-one but not onto done clear
  B)
             Onto but not one-one done clear
  C)
             One-one and onto both done clear
  D)
             Neither one-one nor onto done clear
  View Solution play_arrow

• question_answer46)
  If \[f:[0,\ \infty )\to [0,\ \infty )\] and \[f(x)=\frac{x}{1+x},\]then f is            [IIT Screening 2003]
  A)
             One-one and onto done clear
  B)
             One-one but not onto done clear
  C)
             Onto but not one-one done clear
  D)
             Neither one-one nor onto done clear
  View Solution play_arrow

• question_answer47)
  If \[f:R\to S\] defined by \[f(x)=\sin x-\sqrt{3}\cos x+1\]is onto, then the interval of S is [AIEEE 2004; IIT Screening 2004]
  A)
                     [?1, 3] done clear
  B)
             [1, 1] done clear
  C)
             [0, 1] done clear
  D)
             [0, ?1] done clear
  View Solution play_arrow

• question_answer48)
  If R denotes the set of all real numbers then the function \[f:R\to R\] defined \[f(x)=\ [x]\]                         [Karnataka CET 2004]
  A)
                     One-one only done clear
  B)
             Onto only done clear
  C)
             Both one-one and onto done clear
  D)
             Neither one-one nor onto done clear
  View Solution play_arrow

• question_answer49)
  \[f(x)=x+\sqrt{{{x}^{2}}}\] is a function from R\[\to \]R , then \[f(x)\] is [Orissa JEE 2004]           
  A)
             Injective done clear
  B)
             Surjective done clear
  C)
             Bijective done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer50)
  If \[(x,\,y)\in R\] and \[x,\ y\ne 0\]; \[f(x,\ y)\to \frac{x}{y}\], then this function is a/an [Orissa JEE 2004]
  A)
                     Surjection done clear
  B)
             Bijection done clear
  C)
             One-one done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer51)
  The period of \[f(x)=x-[x]\], if it is periodic, is    [AMU 2000]
  A)
                     \[f(x)\] is not periodic done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             1 done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer52)
  If \[f(x)\] is periodic function with period T then the function \[f(ax+b)\] where \[a>0\], is periodic with period [AMU 2000]
  A)
                     \[T/b\] done clear
  B)
             aT done clear
  C)
             bT done clear
  D)
             \[T/a\] done clear
  View Solution play_arrow

• question_answer53)
  If \[f(x)=ax+b\] and \[g(x)=cx+d\], then \[f(g(x))=g(f(x))\] is equivalent to [UPSEAT 2001]
  A)
             \[f(a)=g(c)\] done clear
  B)
             \[f(b)=g(b)\] done clear
  C)
             \[f(d)=g(b)\] done clear
  D)
             \[f(c)=g(a)\] done clear
  View Solution play_arrow

• question_answer54)
  Domain and range of \[f(x)=\frac{|x-3|}{x-3}\] are respectively
  A)
             \[R,\ [-1,\ 1]\] done clear
  B)
             \[R-\{3\},\ \left\{ 1,\ -1 \right\}\] done clear
  C)
             \[{{R}^{+}},\ R\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer55)
  If in greatest integer function, the domain is a set of real numbers, then range will be set of
  A)
             Real numbers done clear
  B)
             Rational numbers done clear
  C)
             Imaginary numbers done clear
  D)
             Integers done clear
  View Solution play_arrow

• question_answer56)
  Domain of function \[f(x)={{\sin }^{-1}}5x\] is
  A)
             \[\left( -\frac{1}{5},\ \frac{1}{5} \right)\] done clear
  B)
             \[\left[ -\frac{1}{5},\ \frac{1}{5} \right]\] done clear
  C)
             R done clear
  D)
             \[\left( 0,\ \frac{1}{5} \right)\] done clear
  View Solution play_arrow

• question_answer57)
  The domain of the function \[f(x)=\frac{{{\sin }^{-1}}(3-x)}{\ln (|x|\ -2)}\] is [Orissa JEE 2002]
  A)
             [2, 4] done clear
  B)
             (2, 3) È (3, 4] done clear
  C)
             [2,\[\infty \]) done clear
  D)
             \[(-\infty ,\ -3)\cup [2,\ \infty )\] done clear
  View Solution play_arrow

• question_answer58)
  The domain of \[{{\sin }^{-1}}\left[ {{\log }_{3}}\left( \frac{x}{3} \right) \right]\] is [AIEEE 2002]
  A)
             [1, 9] done clear
  B)
             [?1, 9] done clear
  C)
             [?9, 1] done clear
  D)
             [?9, ?1] done clear
  View Solution play_arrow

• question_answer59)
  Domain of the function \[\log |{{x}^{2}}-9|\] is
  A)
             R done clear
  B)
             \[R-[-3,\ 3]\] done clear
  C)
             \[R-\{-3,\ 3\}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer60)
  Domain of \[f(x)=\log |\log x|\] is [DCE 2002]
  A)
             \[(0,\ \infty )\] done clear
  B)
             \[(1,\ \infty )\] done clear
  C)
             \[(0,\ 1)\cup (1,\ \infty )\] done clear
  D)
             \[(-\infty ,\ 1)\] done clear
  View Solution play_arrow

• question_answer61)
   The domain of the function \[f(x)={{\sin }^{-1}}[{{\log }_{2}}(x/2)]\] is [RPET 2002]
  A)
             [1, 4] done clear
  B)
             [?4, 1] done clear
  C)
             [?1, 4] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer62)
  The domain of \[f(x)=\frac{{{\log }_{2}}(x+3)}{{{x}^{2}}+3x+2}\] is [IIT Screening 2001; UPSEAT 2001]
  A)
             \[R-\{-1,\ -2\}\] done clear
  B)
             \[(-2,\ +\infty )\] done clear
  C)
             \[R-\{-1,\ -2,\ -3\}\] done clear
  D)
             \[(-3,\ +\infty )-\{-1,\ -2\}\] done clear
  View Solution play_arrow

• question_answer63)
  The function \[f(x)=\frac{{{\sec }^{-1}}x}{\sqrt{x-[x]}},\] where [.] denotes the greatest integer less than or equal to x is defined for all x belonging to
  A)
             R     done clear
  B)
                     \[R-\{(-1,\ 1)\cup (n|n\in Z)\}\] done clear
  C)
             \[{{R}^{+}}-(0,\ 1)\] done clear
  D)
             \[{{R}^{+}}-\{n|n\in N\}\] done clear
  View Solution play_arrow

• question_answer64)
  If the domain of function \[f(x)={{x}^{2}}-6x+7\] is \[(-\infty ,\ \infty )\], then the range of function is                    [MP PET 1996]
  A)
             \[(-\infty ,\ \infty )\] done clear
  B)
             \[[-2,\ \infty )\] done clear
  C)
             \[(-2,\ 3)\] done clear
  D)
             \[(-\infty ,\ -2)\] done clear
  View Solution play_arrow

• question_answer65)
  The domain of the function \[f(x)=\sqrt{\log \frac{1}{|\sin x|}}\] is [RPET 2001]
  A)
             \[R-\{2n\pi ,\ n\in I\}\] done clear
  B)
             \[R-\{n\pi ,\ n\in I\}\] done clear
  C)
             \[R-\{-\pi ,\ \pi \}\] done clear
  D)
             \[(-\infty ,\ \infty )\] done clear
  View Solution play_arrow

• question_answer66)
  The domain of the function \[f(x)=\log (\sqrt{x-4}+\sqrt{6-x})\] is [RPET 2001]
  A)
             \[[4,\infty )\] done clear
  B)
             \[(-\infty ,\ 6]\] done clear
  C)
             \[[4,\ 6]\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer67)
  . Domain of the function \[f(x)={{\left[ {{\log }_{10}}\left( \frac{5x-{{x}^{2}}}{4} \right) \right]}^{1/2}}\] is [UPSEAT 2001]
  A)
             \[-\infty <x<\infty \] done clear
  B)
             \[1\le x\le 4\] done clear
  C)
             \[4\le x\le 16\] done clear
  D)
             \[-1\le x\le 1\] done clear
  View Solution play_arrow

• question_answer68)
  The domain of the derivative of the function \[f(x)=\left\{ \begin{align}   & {{\tan }^{-1}}x\ \ \ \ \ ,\ |x|\ \le 1 \\  & \frac{1}{2}(|x|\ -1)\ ,\ |x|\ >1 \\ \end{align} \right.\] is [IIT Screening 2002]
  A)
                     \[R-\{0\}\] done clear
  B)
             \[R-\{1\}\] done clear
  C)
             \[R-\{-1\}\] done clear
  D)
             \[R-\{-1,\ 1\}\] done clear
  View Solution play_arrow

• question_answer69)
  The domain of the function \[f(x)={{\log }_{3+x}}({{x}^{2}}-1)\] is [Orissa JEE 2003]           
  A)
             \[(-3,\ -1)\cup (1,\ \infty )\] done clear
  B)
             \[[-3,\ -1)\cup [1,\ \infty )\] done clear
  C)
             \[(-3,\ -2)\cup (-2,\ -1)\cup (1,\ \infty )\]                     done clear
  D)
             \[[-3,\ -2)\cup (-2,\ -1)\cup [1,\ \infty )\] done clear
  View Solution play_arrow

• question_answer70)
  If ?n? is an integer, the domain of the function \[\sqrt{\sin 2x}\] is [MP PET 2003]
  A)
                     \[\left[ n\pi -\frac{\pi }{2},\ n\pi  \right]\] done clear
  B)
             \[\left[ n\pi ,\ n\pi +\frac{\pi }{2} \right]\] done clear
  C)
             \[[(2n-1)\pi ,\ 2n\pi ]\] done clear
  D)
             \[[2n\pi ,\ (2n+1)\pi ]\] done clear
  View Solution play_arrow

• question_answer71)
  Domain of definition of the function \[f(x)=\frac{3}{4-{{x}^{2}}}+{{\log }_{10}}({{x}^{3}}-x)\], is [AIEEE 2003]
  A)
                     (1, 2) done clear
  B)
             \[(-1,\ 0)\cup (1,\ 2)\] done clear
  C)
             \[(1,\ 2)\cup (2,\ \infty )\] done clear
  D)
             \[(-1,\ 0)\cup (1,\ 2)\cup (2,\ \infty )\] done clear
  View Solution play_arrow

• question_answer72)
  Domain of the function \[f(x)=\sqrt{2-2x-{{x}^{2}}}\] is            [BIT Ranchi 1992]           
  A)
             \[-\sqrt{3}\le x\le \sqrt{3}\] done clear
  B)
             \[-1-\sqrt{3}\le x\le -1+\sqrt{3}\] done clear
  C)
             \[-2\le x\le 2\] done clear
  D)
             \[-2+\sqrt{3}\le x\le -2-\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer73)
  Domain of the function \[f(x)=\frac{x-3}{(x-1)\sqrt{{{x}^{2}}-4}}\] is            [BIT Ranchi 1991]           
  A)
             (1, 2) done clear
  B)
             \[(-\infty ,\ -2)\cup (2,\ \infty )\] done clear
  C)
             \[(-\infty ,\ -2)\cup (1,\ \infty )\] done clear
  D)
             \[(-\infty ,\ \infty )-\{1,\ \pm 2\}\] done clear
  View Solution play_arrow

• question_answer74)
  Domain of the function \[\sqrt{\log \left\{ (5x-{{x}^{2}})/6 \right\}}\] is
  A)
                     (2, 3) done clear
  B)
             [2, 3] done clear
  C)
             [1, 2] done clear
  D)
             [1, 3] done clear
  View Solution play_arrow

• question_answer75)
  Domain of the function \[\sqrt{2-x}-\frac{1}{\sqrt{9-{{x}^{2}}}}\] is
  A)
             (?3, 1) done clear
  B)
             [?3, 1] done clear
  C)
             (?3, 2] done clear
  D)
             [?3, 1) done clear
  View Solution play_arrow

• question_answer76)
  Domain of the function \[\frac{\sqrt{1+x}-\sqrt{1-x}}{x}\] is
  A)
             (?1, 1) done clear
  B)
             (?1, 1)?{0} done clear
  C)
             [?1, 1] done clear
  D)
             [?1, 1]?{0} done clear
  View Solution play_arrow

• question_answer77)
  The domain of the function \[f(x)=\sqrt{x-{{x}^{2}}}+\sqrt{4+x}+\sqrt{4-x}\] is [AMU 1999]
  A)
                     \[[-4,\ \infty )\] done clear
  B)
              [?4, 4] done clear
  C)
             [0, 4] done clear
  D)
             [0, 1] done clear
  View Solution play_arrow

• question_answer78)
  The domain of the function \[f(x)={{\sin }^{-1}}\{{{(1+{{e}^{x}})}^{-1}}\}\] is                                                               [AMU 1999]           
  A)
             \[\left( \frac{1}{4},\ \frac{1}{3} \right)\] done clear
  B)
              [?1, 0] done clear
  C)
             [0, 1] done clear
  D)
             [?1, 1] done clear
  View Solution play_arrow

• question_answer79)
  The domain of the function \[\sqrt{\log ({{x}^{2}}-6x+6)}\] is            [Roorkee 1999; MP PET 2002]
  A)
             \[(-\infty ,\ \infty )\]                     done clear
  B)
             \[(-\infty ,\ 3-\sqrt{3})\cup (3+\sqrt{3},\ \infty )\] done clear
  C)
             \[(-\infty ,\ 1]\cup [5,\ \infty )\]                                       done clear
  D)
             \[[0,\ \infty )\] done clear
  View Solution play_arrow

• question_answer80)
  The largest possible set of real numbers which can be the domain of \[f(x)=\sqrt{1-\frac{1}{x}}\] is                          [AMU 2000]
  A)
                     \[(0,\ 1)\cup (0,\ \infty )\] done clear
  B)
              \[(-1,\ 0)\cup (1,\ \infty )\] done clear
  C)
             \[(-\infty ,\ -1)\cup (0,\ \infty )\] done clear
  D)
             \[(-\infty ,\ 0)\cup (1,\ \infty )\] done clear
  View Solution play_arrow

• question_answer81)
   Domain of the function \[f(x)={{\sin }^{-1}}(1+3x+2{{x}^{2}})\] is [Roorkee 2000]           
  A)
             \[(-\infty ,\ \infty )\] done clear
  B)
              \[(-1,\ 1)\] done clear
  C)
             \[\left[ -\frac{3}{2},\ 0 \right]\] done clear
  D)
             \[\left( -\infty ,\ \frac{-1}{2} \right)\cup (2,\ \infty )\] done clear
  View Solution play_arrow

• question_answer82)
  Domain of the function \[f(x)=\frac{{{x}^{2}}-3x+2}{{{x}^{2}}+x-6}\] is
  A)
                     \[\{x:x\in R,\ \ x\ne 3\}\] done clear
  B)
                     \[\{x:x\in R,\ \ x\ne 2\}\] done clear
  C)
             \[\{x:x\in R\}\] done clear
  D)
             \[\{x:x\in R,\ \ x\ne 2,\ x\ne -3\}\] done clear
  View Solution play_arrow

• question_answer83)
  Domain of \[f(x)={{({{x}^{2}}-1)}^{-1/2}}\] is [Roorkee 1987]           
  A)
             \[(-\infty ,\ -1)\cup (1,\ \infty )\] done clear
  B)
              \[(-\infty ,\ -1]\cup (1,\ \infty )\] done clear
  C)
             \[(-\infty ,\ -1]\cup [1,\ \infty )\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer84)
  The domain of the function \[y=\frac{1}{\sqrt{|x|\ -x}}\] is              [Roorkee 1998; RPET 2000]           
  A)
             \[(-\infty ,\ 0)\] done clear
  B)
              \[(-\infty ,\ 0]\] done clear
  C)
             \[(-\infty ,\ -1)\] done clear
  D)
             \[(-\infty ,\ \infty )\] done clear
  View Solution play_arrow

• question_answer85)
  The natural domain of the real valued function defined by \[f(x)=\sqrt{{{x}^{2}}-1}+\sqrt{{{x}^{2}}+1}\] is                        [SCRA 1996]
  A)
                     \[1<x<\infty \] done clear
  B)
              \[-\infty <x<\infty \] done clear
  C)
             \[-\infty <x<-1\] done clear
  D)
             \[(-\infty ,\ \infty )-(-1,\ 1)\] done clear
  View Solution play_arrow

• question_answer86)
  The domain of the function \[f(x)=\exp (\sqrt{5x-3-2{{x}^{2}}})\] is  [MP PET 2004]           
  A)
             \[\left[ 1,\ -\frac{3}{2} \right]\] done clear
  B)
              \[\left[ \frac{3}{2},\ \infty  \right]\] done clear
  C)
             \[[-\infty ,\ 1]\] done clear
  D)
             \[\left[ 1,\ \frac{3}{2} \right]\] done clear
  View Solution play_arrow

• question_answer87)
  The domain of the function \[f(x)=\frac{{{\sin }^{-1}}(x-3)}{\sqrt{9-{{x}^{2}}}}\] is            [AIEEE 2004]           
  A)
             [1, 2) done clear
  B)
             [2, 3) done clear
  C)
             [1, 2] done clear
  D)
             [2, 3] done clear
  View Solution play_arrow

• question_answer88)
  The range of \[f(x)=\sec \left( \frac{\pi }{4}{{\cos }^{2}}x \right)\,,\ -\infty <x<\infty \] is [Orissa JEE 2002]           
  A)
             \[[1,\ \sqrt{2}]\] done clear
  B)
              \[[1,\ \infty )\] done clear
  C)
             \[[-\sqrt{2},\ -1]\cup [1,\ \sqrt{2}]\] done clear
  D)
             \[(-\infty ,\ -1]\cup [1,\ \infty )\] done clear
  View Solution play_arrow

• question_answer89)
  Range of the function \[f(x)=\frac{{{x}^{2}}+x+2}{{{x}^{2}}+x+1};\ x\in R\] is            [IIT Screening 2003]           
  A)
             \[(1,\ \infty )\] done clear
  B)
              \[(1,\ 11/7]\] done clear
  C)
             \[(1,\ 7/3]\] done clear
  D)
             \[(1,\ 7/5]\] done clear
  View Solution play_arrow

• question_answer90)
  If \[f(x)=a\cos (bx+c)+d\], then range of \[f(x)\] is            [UPSEAT 2001]
  A)
                     \[[d+a,\ d+2a]\] done clear
  B)
              \[[a-d,\ a+d]\] done clear
  C)
             \[[d+a,\ a-d]\] done clear
  D)
             \[[d-a,\ d+a]\] done clear
  View Solution play_arrow

• question_answer91)
  Range of \[f(x)=\ [x]\ -x\] is
  A)
                     [0, 1] done clear
  B)
              (?1, 0] done clear
  C)
             R done clear
  D)
             (?1, 1) done clear
  View Solution play_arrow

• question_answer92)
  The range of \[f(x)=\cos (x/3)\] is [RPET 2002]
  A)
                     \[(-1/3,\ 1/3)\] done clear
  B)
              \[[-1,\ 1]\] done clear
  C)
             \[(1/3,\ -1/3)\] done clear
  D)
             \[(-3,\ 3)\] done clear
  View Solution play_arrow

• question_answer93)
  The range of the function \[f(x)=\frac{x+2}{|x+2|}\] is [RPET 2002]
  A)
             {0, 1} done clear
  B)
              {?1, 1} done clear
  C)
             R done clear
  D)
             \[R-\{-2\}\] done clear
  View Solution play_arrow

• question_answer94)
  The range of \[f(x)=\cos x-\sin x\] is            [MP PET 1995; Pb. CET 2001]           
  A)
             \[(-1,\ 1)\] done clear
  B)
              \[[-1,\,\ 1)\] done clear
  C)
             \[\left[ -\frac{\pi }{2},\ \frac{\pi }{2} \right]\] done clear
  D)
             \[[-\sqrt{2},\ \sqrt{2}]\] done clear
  View Solution play_arrow

• question_answer95)
  If \[f:R\to R\], then the range of the function \[f(x)=\frac{{{x}^{2}}}{{{x}^{2}}+1}\] is [MP PET 1987]
  A)
                     \[{{R}^{-}}\] done clear
  B)
             \[{{R}^{+}}\] done clear
  C)
             R done clear
  D)
             \[R\times R\] done clear
  View Solution play_arrow

• question_answer96)
  The range of \[f(x)=\cos 2x-\sin 2x\] contains the set [IIT Screening]
  A)
                     [2, 4] done clear
  B)
             [?1, 1] done clear
  C)
             [?2, 2] done clear
  D)
             [?4, 4] done clear
  View Solution play_arrow

• question_answer97)
  Range of the function \[\frac{1}{2-\sin 3x}\] is [AMU 1999]
  A)
                     [1, 3] done clear
  B)
              \[\left[ \frac{1}{3},\,\,1 \right]\] done clear
  C)
             (1, 3) done clear
  D)
             \[\left( \frac{1}{3},\ 1 \right)\] done clear
  View Solution play_arrow

• question_answer98)
  Range of the function \[f(x)={{\sin }^{2}}({{x}^{4}})+{{\cos }^{2}}({{x}^{4}})\] is
  A)
                     \[(-\infty ,\ \infty )\] done clear
  B)
              {1} done clear
  C)
             (?1, 1) done clear
  D)
             (0, 1) done clear
  View Solution play_arrow

• question_answer99)
  Range of the function \[f(x)=9-7\sin x\] is
  A)
             (2, 16) done clear
  B)
             [2, 16] done clear
  C)
             [?1, 1] done clear
  D)
             (2, 16] done clear
  View Solution play_arrow

• question_answer100)
  Range of \[f(x)=\frac{{{x}^{2}}+34x-71}{{{x}^{2}}+2x-7}\] is [Roorkee 1983]
  A)
                     [5, 9] done clear
  B)
             \[(-\infty ,\ 5]\cup [9,\ \infty )\] done clear
  C)
             (5, 9) done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer101)
  The interval for which \[{{\sin }^{-1}}\sqrt{x}+{{\cos }^{-1}}\sqrt{x}=\frac{\pi }{2}\] holds [IIT Screening]
  A)
                     \[[0,\ \infty )\] done clear
  B)
             \[[0,\ 3]\] done clear
  C)
             [0, 1] done clear
  D)
             [0, 2] done clear
  View Solution play_arrow

• question_answer102)
  Function \[{{\sin }^{-1}}\sqrt{x}\] is defined in the interval
  A)
                     (?1, 1) done clear
  B)
             [0, 1] done clear
  C)
             [?1, 0] done clear
  D)
             (?1, 2) done clear
  View Solution play_arrow

• question_answer103)
  The function \[f:R\to R\] is defined by \[f(x)={{\cos }^{2}}x+{{\sin }^{4}}x\] for \[x\in R\], then \[f(R)=\]            [EAMCET 2002]
  A)
                     \[\left( \frac{3}{4},\ 1 \right]\] done clear
  B)
              \[\left[ \frac{3}{4},\ 1 \right)\] done clear
  C)
             \[\left[ \frac{3}{4},\ 1 \right]\] done clear
  D)
             \[\left( \frac{3}{4},\ 1 \right)\] done clear
  View Solution play_arrow

• question_answer104)
  If x is real, then value of the expression \[\frac{{{x}^{2}}+14x+9}{{{x}^{2}}+2x+3}\] lies between [UPSEAT 2002]
  A)
                     5 and 4 done clear
  B)
             5 and ?4 done clear
  C)
             ? 5 and 4 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer105)
  For \[\theta >\frac{\pi }{3}\], the value of \[f(\theta )={{\sec }^{2}}\theta +{{\cos }^{2}}\theta \] always lies in the interval [Orissa JEE 2002]
  A)
                     (0, 2) done clear
  B)
             [0, 1] done clear
  C)
             (1, 2) done clear
  D)
             \[[2,\ \infty )\] done clear
  View Solution play_arrow

• question_answer106)
  Which of the following function is even function            [RPET 2000]           
  A)
             \[f(x)=\frac{{{a}^{x}}+1}{{{a}^{x}}-1}\] done clear
  B)
             \[f(x)=x\left( \frac{{{a}^{x}}-1}{{{a}^{x}}+1} \right)\] done clear
  C)
             \[f(x)=\frac{{{a}^{x}}-{{a}^{-x}}}{{{a}^{x}}+{{a}^{-x}}}\] done clear
  D)
             \[f(x)=\sin x\] done clear
  View Solution play_arrow

• question_answer107)
  If \[f(x)=\log \frac{1+x}{1-x}\], then \[f(x)\] is [Kerala (Engg.) 2002]
  A)
                     Even function done clear
  B)
             \[f({{x}_{1}})f({{x}_{2}})=f({{x}_{1}}+{{x}_{2}})\] done clear
  C)
                     \[\frac{f({{x}_{1}})}{f({{x}_{2}})}=f({{x}_{1}}-{{x}_{2}})\] done clear
  D)
             Odd function done clear
  View Solution play_arrow

• question_answer108)
  The function \[f(x)=\sin \left( \log (x+\sqrt{{{x}^{2}}+1}) \right)\] is            [Orissa JEE 2002]
  A)
                     Even function done clear
  B)
             Odd function done clear
  C)
                     Neither even nor odd done clear
  D)
              Periodic function done clear
  View Solution play_arrow

• question_answer109)
  The function \[f(x)=\log (x+\sqrt{{{x}^{2}}+1})\], is            [AIEEE 2003; MP PET 2003; UPSEAT 2003]
  A)
             An even function                done clear
  B)
             An odd function done clear
  C)
             A Periodic function             done clear
  D)
             Neither an even nor odd function done clear
  View Solution play_arrow

• question_answer110)
  Which of the following function is invertible      [AMU 2001]
  A)
                     \[f(x)={{2}^{x}}\] done clear
  B)
             \[f(x)={{x}^{3}}-x\] done clear
  C)
             \[f(x)={{x}^{2}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer111)
  If \[y=f(x)=\frac{x+2}{x-1}\], then \[x=\] [IIT 1984]
  A)
                     \[f(y)\]                             done clear
  B)
             \[2f(y)\] done clear
  C)
                     \[\frac{1}{f(y)}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer112)
  Which of the following functions is inverse of itself
  A)
                     \[f(x)=\frac{1-x}{1+x}\] done clear
  B)
             \[f(x)={{5}^{\log x}}\] done clear
  C)
                     \[f(x)={{2}^{x(x-1)}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer113)
  The inverse of the function \[f(x)=\frac{{{e}^{x}}-{{e}^{-x}}}{{{e}^{x}}+{{e}^{-x}}}+2\] is given by [Kurukshetra CEE 1996]
  A)
                     \[{{\log }_{e}}{{\left( \frac{x-2}{x-1} \right)}^{1/2}}\] done clear
  B)
             \[{{\log }_{e}}{{\left( \frac{x-1}{3-x} \right)}^{1/2}}\] done clear
  C)
                     \[{{\log }_{e}}{{\left( \frac{x}{2-x} \right)}^{1/2}}\] done clear
  D)
             \[{{\log }_{e}}{{\left( \frac{x-1}{x+1} \right)}^{-2}}\] done clear
  View Solution play_arrow

• question_answer114)
  If the function \[f:[1,\ \infty )\to [1,\ \infty )\] is defined by \[f(x)={{2}^{x(x-1)}},\] then \[{{f}^{-1}}\](x) is [IIT 1999]
  A)
                     \[{{\left( \frac{1}{2} \right)}^{x(x-1)}}\] done clear
  B)
             \[\frac{1}{2}(1+\sqrt{1+4{{\log }_{2}}x})\] done clear
  C)
                     \[\frac{1}{2}(1-\sqrt{1+4{{\log }_{2}}x})\] done clear
  D)
             Not defined done clear
  View Solution play_arrow

• question_answer115)
  If \[f(x)=3x-5\], then \[{{f}^{-1}}(x)\] [IIT 1998]
  A)
                     Is given by \[\frac{1}{3x-5}\]                                      done clear
  B)
                     Is given by \[\frac{x+5}{3}\] done clear
  C)
                     Does not exist because f  is not one-one done clear
  D)
                     Does not exist because f is not onto done clear
  View Solution play_arrow

• question_answer116)
  If \[f:IR\to IR\] is defined by \[f(x)=3x-4\], then \[{{f}^{-1}}:IR\to IR\] is [SCRA 1996]
  A)
                     \[4-3x\] done clear
  B)
             \[\frac{x+4}{3}\] done clear
  C)
                     \[\frac{1}{3x-4}\] done clear
  D)
             \[\frac{3}{x+4}\] done clear
  View Solution play_arrow

• question_answer117)
  If \[f(x)=\frac{x}{1+x}\], then \[{{f}^{-1}}(x)\] is equal to [AMU 1999]
  A)
                     \[\frac{(1+x)}{x}\] done clear
  B)
             \[\frac{1}{(1+x)}\] done clear
  C)
                     \[\frac{(1+x)}{(1-x)}\] done clear
  D)
             \[\frac{x}{(1-x)}\] done clear
  View Solution play_arrow

• question_answer118)
  Which of the following function is inverse function            [AMU 2000]
  A)
             \[f(x)=\frac{1}{x-1}\] done clear
  B)
             \[f(x)={{x}^{2}}\] for all\[x\] done clear
  C)
             \[f(x)={{x}^{2}}\], \[x\ge 0\] done clear
  D)
             \[f(x)={{x}^{2}},\ x\le 0\] done clear
  View Solution play_arrow

• question_answer119)
  Let \[f(\theta )=\sin \theta (\sin \theta +\sin 3\theta )\], then \[f(\theta )\]            [IIT Screening 2000]
  A)
             \[\ge 0\] only when \[\theta \ge 0\] done clear
  B)
             \[\le 0\] for all real \[\theta \] done clear
  C)
                     \[\ge 0\] for all real \[\theta \] done clear
  D)
             \[\le 0\]only when \[\theta \le 0\] done clear
  View Solution play_arrow

• question_answer120)
  The inverse of the function \[\frac{{{10}^{x}}-{{10}^{-x}}}{{{10}^{x}}+{{10}^{-x}}}\] is      [RPET 2001]
  A)
                     \[\frac{1}{2}{{\log }_{10}}\left( \frac{1+x}{1-x} \right)\] done clear
  B)
             \[\frac{1}{2}{{\log }_{10}}\left( \frac{1-x}{1+x} \right)\] done clear
  C)
                     \[\frac{1}{4}{{\log }_{10}}\left( \frac{2x}{2-x} \right)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer121)
  Inverse of the function \[y=2x-3\] is [UPSEAT 2002]
  A)
                     \[\frac{x+3}{2}\] done clear
  B)
             \[\frac{x-3}{2}\] done clear
  C)
                     \[\frac{1}{2x-3}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer122)
  Let the function f be defined by \[f(x)=\frac{2x+1}{1-3x}\], then \[{{f}^{-1}}(x)\] is [Kerala (Engg.) 2002]
  A)
                     \[\frac{x-1}{3x+2}\] done clear
  B)
             \[\frac{3x+2}{x-1}\] done clear
  C)
                     \[\frac{x+1}{3x-2}\] done clear
  D)
             \[\frac{2x+1}{1-3x}\] done clear
  View Solution play_arrow

• question_answer123)
  If \[f(x)={{x}^{2}}+1\], then \[{{f}^{-1}}(17)\] and \[{{f}^{-1}}(-3)\]will be [UPSEAT 2003]
  A)
                     4, 1 done clear
  B)
             4, 0 done clear
  C)
                     3, 2 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer124)
  Let \[f(x)=\sin x+\cos x,\ g(x)={{x}^{2}}-1\]. Thus \[g(f(x))\] is invertible for \[x\in \] [IIT Screening 2004]
  A)
                     \[\left[ -\frac{\pi }{2},\ 0 \right]\] done clear
  B)
             \[\left[ -\frac{\pi }{2},\ \pi  \right]\] done clear
  C)
                     \[\left[ -\frac{\pi }{2},\ \frac{\pi }{4} \right]\] done clear
  D)
             \[\left[ 0,\ \frac{\pi }{2} \right]\] done clear
  View Solution play_arrow

• question_answer125)
  If \[f(x)=\frac{2x-1}{x+5}\]\[(x\ne -5)\], then \[{{f}^{-1}}(x)\] is equal to            [MP PET 2004]
  A)
                     \[\frac{x+5}{2x-1},\ x\ne \frac{1}{2}\] done clear
  B)
             \[\frac{5x+1}{2-x},\ x\ne 2\] done clear
  C)
                     \[\frac{5x-1}{2-x},\ x\ne 2\] done clear
  D)
             \[\frac{x-5}{2x+1},\ x\ne \frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer126)
  If f be the greatest integer function and g be the modulus function, then \[(gof)\left( -\frac{5}{3} \right)-(fog)\left( -\frac{5}{3} \right)=\]
  A)
                     1 done clear
  B)
             ?1 done clear
  C)
                     2 done clear
  D)
             4 done clear
  View Solution play_arrow

• question_answer127)
  If \[f(x)=2x\] and g is identity function, then
  A)
                     \[(fog)(x)=g(x)\] done clear
  B)
             \[(g+g)(x)=g(x)\] done clear
  C)
                     \[(fog)(x)=(g+g)(x)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer128)
  If \[f(x)={{x}^{2}}-1\] and \[g(x)=3x+1\], then \[(gof)(x)=\]
  A)
                     \[{{x}^{2}}-1\] done clear
  B)
             \[2{{x}^{2}}-1\] done clear
  C)
                     \[3{{x}^{2}}-2\] done clear
  D)
             \[2{{x}^{2}}+2\] done clear
  View Solution play_arrow

• question_answer129)
  If f is an exponential function and g is a logarithmic function, then \[fog(1)\] will be
  A)
                     e done clear
  B)
             \[{{\log }_{e}}e\] done clear
  C)
                     0 done clear
  D)
             2e done clear
  View Solution play_arrow

• question_answer130)
  If \[f(x)={{e}^{2x}}\] and \[g(x)=\log \sqrt{x}\]\[(x>0)\], then \[fog(x)\] is equal to
  A)
                     \[{{e}^{2x}}\] done clear
  B)
             \[\log \sqrt{x}\] done clear
  C)
                     \[{{e}^{2x}}\log \sqrt{x}\] done clear
  D)
             x done clear
  View Solution play_arrow

• question_answer131)
  If \[f(x)=|\cos x|\]and \[g(x)=[x]\], then \[gof(x)\] is equal to
  A)
                     \[|\cos \ [x]|\] done clear
  B)
             \[|\cos x|\] done clear
  C)
                     \[[|\cos x|]\] done clear
  D)
             \[|[\cos x]|\] done clear
  View Solution play_arrow

• question_answer132)
  If \[f(x)={{x}^{2}}+1\],then \[fof(x)\] is equal to
  A)
                     \[{{x}^{2}}+1\] done clear
  B)
             \[{{x}^{2}}+2x+2\] done clear
  C)
                     \[{{x}^{4}}+2{{x}^{2}}+2\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer133)
  If \[f(x)=\frac{x}{\sqrt{1+{{x}^{2}}}}\], then \[(fofof)(x)=\]           [RPET 2000]
  A)
                     \[\frac{3x}{\sqrt{1+{{x}^{2}}}}\] done clear
  B)
             \[\frac{x}{\sqrt{1+3{{x}^{2}}}}\] done clear
  C)
                     \[\frac{3x}{\sqrt{1+{{x}^{2}}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer134)
  If \[\varphi (x)={{x}^{2}}+1\] and \[\psi (x)={{3}^{x}}\], then \[\varphi \{\psi (x)\}\] and \[\psi \{\varphi (x)\}=\]
  A)
                     \[{{3}^{2x+1}},\ {{3}^{{{x}^{2}}+1}}\] done clear
  B)
             \[{{3}^{2x+1}},\ {{3}^{{{x}^{2}}}}+1\] done clear
  C)
                     \[{{3}^{2x}}+1,\ {{3}^{{{x}^{2}}+1}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer135)
  If \[g(x)={{x}^{2}}+x-2\] and \[\frac{1}{2}gof(x)=2{{x}^{2}}-5x+2\], then \[f(x)\] is [Roorkee 1998; MP PET 2002]
  A)
                     \[2x-3\] done clear
  B)
             \[2x+3\] done clear
  C)
                     \[2{{x}^{2}}+3x+1\] done clear
  D)
             \[2{{x}^{2}}-3x-1\] done clear
  View Solution play_arrow

• question_answer136)
  If \[f(x)={{\log }_{a}}x\] and \[F(x)={{a}^{x}}\], then \[F[f(x)]\] is            [SCRA 1996]
  A)
                     \[f[F(x)]\] done clear
  B)
             \[f[F(2x)]\] done clear
  C)
                     \[F|f(2x)|\] done clear
  D)
             \[F[(x)]\] done clear
  View Solution play_arrow

• question_answer137)
  Let f and g be functions defined by \[f(x)=\frac{x}{x+1},\]\[g(x)=\frac{x}{1-x}\], then \[(fog)(x)\] is         [SCRA 1996]
  A)
                     \[\frac{1}{x}\] done clear
  B)
             \[\frac{1}{x-1}\] done clear
  C)
                     \[x-1\] done clear
  D)
             x done clear
  View Solution play_arrow

• question_answer138)
  If from \[R\to R\], \[f(x)={{(x+1)}^{2}}\], \[g(x)={{x}^{2}}+1\], then \[(fog)(-3)\] equals [RPET 1999]
  A)
                     121 done clear
  B)
             112 done clear
  C)
                     211 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer139)
  Suppose that \[g(x)=1+\sqrt{x}\] and \[f(g(x))=3+2\sqrt{x}+x\], then \[f(x)\] is [MP PET 2000; Karnataka CET 2002]
  A)
                     \[1+2{{x}^{2}}\] done clear
  B)
             \[2+{{x}^{2}}\] done clear
  C)
                     \[1+x\] done clear
  D)
             \[2+x\] done clear
  View Solution play_arrow

• question_answer140)
  The composite mapping \[fog\]of the map \[f:R\to R\], \[f(x)=\sin x\], \[g:R\to R\], \[g(x)={{x}^{2}}\]is [UPSEAT 2000]
  A)
                     \[\sin x+{{x}^{2}}\] done clear
  B)
             \[{{(\sin x)}^{2}}\] done clear
  C)
                     \[\sin {{x}^{2}}\] done clear
  D)
             \[\frac{\sin x}{{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer141)
  Let \[f(x)=ax+b\] and \[g(x)=cx+d,\ a\ne 0,\ c\ne 0\]. Assume \[a=1,\ b=2\]. If \[(fog)(x)=(gof)(x)\] for all x, what can you say about c and d [AMU 2000]
  A)
                     c and d both arbitrary done clear
  B)
             \[c=1,\ d\] arbitrary done clear
  C)
             c arbitrary, \[d=1\] done clear
  D)
             \[c=1,\ d=1\] done clear
  View Solution play_arrow

• question_answer142)
  Let \[g(x)=1+x-[x]\] and \[f(x)=\left\{ \begin{align}   & -1,\ x<0 \\  & 0,\ \ x=0,\  \\  & \text{1,}\ \ \ \text{x}>\text{0} \\ \end{align} \right.\]then for all \[x,\ f(g(x))\] is equal to [IIT Screening 2001; UPSEAT 2001]
  A)
                     x done clear
  B)
             1 done clear
  C)
                     \[f(x)\] done clear
  D)
             \[g(x)\] done clear
  View Solution play_arrow

• question_answer143)
  If \[f(x)=\frac{\alpha \,x}{x+1},\ x\ne -1\]. Then, for what value of \[\alpha \] is \[f(f(x))=x\] [IIT Screening 2001; UPSEAT 2001]
  A)
                     \[\sqrt{2}\] done clear
  B)
             \[-\sqrt{2}\] done clear
  C)
                     1 done clear
  D)
             ?1 done clear
  View Solution play_arrow

• question_answer144)
  If \[f(x)=\frac{2x+1}{3x-2}\], then \[(fof)(2)\] is equal to            [Kerala (Engg.) 2002]
  A)
                     1 done clear
  B)
             3 done clear
  C)
                     4 done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer145)
  If \[f(x)={{\sin }^{2}}x\] and the composite function \[g\{f(x)\}=|\sin x|\], then the function \[g(x)\] is equal to            [Orissa JEE 2003]
  A)
                     \[\sqrt{x-1}\] done clear
  B)
             \[\sqrt{x}\] done clear
  C)
                     \[\sqrt{x+1}\] done clear
  D)
             \[-\sqrt{x}\] done clear
  View Solution play_arrow

• question_answer146)
  If \[f(x)={{(a-{{x}^{n}})}^{1/n}},\]where \[a>0\]and n is a positive integer, then \[f[f(x)]=\]         [IIT 1983; UPSEAT 2001, 04]
  A)
             \[{{x}^{3}}\] done clear
  B)
             \[{{x}^{2}}\] done clear
  C)
             \[x\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer147)
  Let \[f:(-1,1)\to B\], be a function defined by \[f(x)={{\tan }^{-1}}\frac{2x}{1-{{x}^{2}}},\] then f is both one- one and onto when B is the interval [AIEEE 2005]
  A)
             \[\left[ -\frac{\pi }{2},\frac{\pi }{2} \right]\] done clear
  B)
             \[\left( -\frac{\pi }{2},\frac{\pi }{2} \right)\] done clear
  C)
             \[\left( 0,\frac{\pi }{2} \right)\] done clear
  D)
             \[\left[ 0,\frac{\pi }{2} \right)\] done clear
  View Solution play_arrow

• question_answer148)
  A real valued function \[f(x)\] satisfies the function equation \[f(x-y)=f(x)f(y)-f(a-x)f(a+y)\] where a is a given constant and \[f(0)=1\], \[f(2a-x)\] is equal to                                         [AIEEE 2005]
  A)
                     \[f(a)+f(a-x)\] done clear
  B)
             \[f(-x)\] done clear
  C)
             \[-f(x)\] done clear
  D)
             \[f(x)\] done clear
  View Solution play_arrow

• question_answer149)
  If X and Y are two non- empty sets where \[f:X\to Y\]is function  is defined such that \[f(c)=\left\{ f(x):x\in C \right\}\]for \[C\subseteq X\]and \[{{f}^{-1}}(D)=\{x:f(x)\in D\}\]for \[D\subseteq Y\] for any \[A\subseteq X\] and \[B\subseteq Y,\]then                                                                     [IIT Screening 2005]
  A)
                     \[{{f}^{-1}}(f(A))=A\] done clear
  B)
                     \[{{f}^{-1}}(f(A))=A\]only if \[f(x)=Y\] done clear
  C)
             \[f({{f}^{-1}}(B))=B\] only if \[B\subseteq f(X)\] done clear
  D)
             \[f({{f}^{-1}}(B))=B\] done clear
  View Solution play_arrow

• question_answer150)
  If \[f(x)=2{{x}^{6}}+3{{x}^{4}}+4{{x}^{2}}\] then \[f'(x)\] is           [DCE 2005]
  A)
             Even function done clear
  B)
             An odd function done clear
  C)
             Neither even nor odd done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer151)
  If \[f(x)=\frac{\alpha x}{x+1},x\ne -1\], for what value of \[\alpha \] is \[f(f(x))=x\] [Kerala (Engg.) 2005]
  A)
             \[\sqrt{2}\] done clear
  B)
             \[-\sqrt{2}\] done clear
  C)
             1 done clear
  D)
             2 done clear
  E)
             ?1 done clear
  View Solution play_arrow

• question_answer152)
  Function \[f(x)=x-[\,],\] where [ ] shows a greatest integer. This function is                                         [DCE 2005]
  A)
             A periodic function done clear
  B)
             A periodic function whose period is \[\frac{1}{2}\] done clear
  C)
             A periodic function whose period is 1 done clear
  D)
             Not a periodic function done clear
  View Solution play_arrow

• question_answer153)
  Let \[g(x)=1+x-[x]\] and \[f(x)=\left\{ \begin{align}   & -1,\,\,\,If\,\,x<0 \\  & \,\,0,\,\,\,If\,\,\,x=0, \\  & \,\,1,\,\,\,\,\,if\,\,\,\,x>0 \\ \end{align} \right.\]then for all values of x the value of \[fog(x)\]                                             [DCE 2005]
  A)
                     x done clear
  B)
             1 done clear
  C)
                     \[f(x)\] done clear
  D)
              \[g(x)\] done clear
  View Solution play_arrow

• question_answer154)
  If  \[g:[-2,\,2]\to R\]where \[g(x)=\] \[{{x}^{3}}+\tan x+\left[ \frac{{{x}^{2}}+1}{P} \right]\]is a odd function then the value of parametric P is            [DCE 2005]
  A)
                     \[-5<P<5\] done clear
  B)
             \[P<5\] done clear
  C)
                     \[P>5\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer155)
  The Domain of function \[f(x)={{\log }_{e}}(x-[x])\] is            [AMU 2005]
  A)
             R done clear
  B)
             R-Z done clear
  C)
             \[(0,+\infty )\] done clear
  D)
             Z done clear
  View Solution play_arrow

• question_answer156)
  The domain of \[{{\sin }^{-1}}({{\log }_{3}}x)\] is [Kerala (Engg.) 2005]
  A)
             [?1, 1] done clear
  B)
             [0, 1] done clear
  C)
             [0, \[\infty \]] done clear
  D)
             R done clear
  E)
             [1/3, 3] done clear
  View Solution play_arrow

• question_answer157)
  If \[f({{x}_{1}})-f({{x}_{2}})=f\left( \frac{{{x}_{1}}-{{x}_{2}}}{1-{{x}_{1}}{{x}_{2}}} \right)\] for \[{{x}_{1}},{{x}_{2}}\in [-1,\,1]\], then \[f(x)\] is                [Roorkee 1998]
  A)
                     \[\log \frac{(1-x)}{(1+x)}\] done clear
  B)
             \[{{\tan }^{-1}}\frac{(1-x)}{(1+x)}\] done clear
  C)
                     \[\log \frac{(1+x)}{(1-x)}\] done clear
  D)
             \[{{\tan }^{-1}}\frac{(1+x)}{(1-x)}\] done clear
  View Solution play_arrow

• question_answer158)
  If equation of the curve remain unchanged by replacing x and y from ?x and ?y respectively, then the curve is
  A)
                     Symmetric along the x-axis done clear
  B)
                     Symmetric along the y-axis done clear
  C)
             Symmetric in opposite quadrants done clear
  D)
             Symmetric along the line y =x done clear
  View Solution play_arrow

• question_answer159)
  If equation of the curve remain unchanged by replacing x and y from y and x  respectively, then the curve is
  A)
                     Symmetric along  x-axis            done clear
  B)
             Symmetric along y-axis done clear
  C)
             Symmetric along the line y = ? x done clear
  D)
             Symmetric along the line y = x done clear
  View Solution play_arrow

• question_answer160)
  A condition for a function \[y=f(x)\] to have an inverse is that it should be
  A)
                     Defined for all x done clear
  B)
                     Continuous everywhere done clear
  C)
             Strictly monotonic and continuous in the domain            done clear
  D)
             An even function done clear
  View Solution play_arrow

• question_answer161)
  If\[f(x)=\left\{ \begin{align}   & x,\,\,\text{when}\,x\,\text{is}\,\text{rational} \\  & 0\text{,}\,\,\text{when }x\text{ is irrational} \\ \end{align} \right.\]; \[g(x)=\left\{ \begin{align}   & 0,\,\,\,\,\text{when}\,x\,\text{is}\,\text{rational} \\  & x,\,\,\,\,\text{when}\,x\,\text{is irrational} \\ \end{align} \right.\] then \[(f-g)\] is [IIT Screening 2005]
  A)
                     One-one onto done clear
  B)
                     One-one not onto done clear
  C)
                     Not one-one but onto      done clear
  D)
             Not one-one not onto done clear
  View Solution play_arrow

• question_answer162)
  Range of the function \[f(x)=\frac{{{x}^{2}}}{{{x}^{2}}+1}\]is [Orissa JEE 2005]
  A)
             (?1, 0) done clear
  B)
             (?1, 1) done clear
  C)
             [0, 1) done clear
  D)
             (1, 1) done clear
  View Solution play_arrow

• question_answer163)
  The function f satisfies the functional equation \[3f(x)+2f\left( \frac{x+59}{x-1} \right)=10x+30\] for all real \[x\ne 1\]. The value of \[f(7)\] is [Kerala (Engg.) 2005]
  A)
             8 done clear
  B)
             4 done clear
  C)
             ?8 done clear
  D)
             11 done clear
  E)
             44 done clear
  View Solution play_arrow

• question_answer164)
  If \[{{e}^{x}}=y+\sqrt{1+{{y}^{2}}}\], then y = [MNR 1990, UPSEAT 2000]
  A)
             \[\frac{{{e}^{x}}+{{e}^{-x}}}{2}\] done clear
  B)
             \[\frac{{{e}^{x}}-{{e}^{-x}}}{2}\] done clear
  C)
             \[{{e}^{x}}+{{e}^{-x}}\] done clear
  D)
             \[{{e}^{x}}-{{e}^{-x}}\] done clear
  View Solution play_arrow

• question_answer165)
  Let \[f:(2,\,3)\to (0,\,1)\] be defined by \[f(x)=x-[x]\] then \[{{f}^{-1}}(x)\] equals [Orissa JEE 2005]
  A)
             \[x-2\] done clear
  B)
             \[x+1\] done clear
  C)
             \[x-1\] done clear
  D)
             \[x+2\] done clear
  View Solution play_arrow