A) \[{{\left( \frac{x+7}{2} \right)}^{1/3}}\]
B) \[{{\left( x-\frac{7}{2} \right)}^{1/3}}\]
C) \[{{\left( \frac{x-2}{7} \right)}^{1/3}}\]
D) \[{{\left( \frac{x-7}{2} \right)}^{1/3}}\]
Correct Answer: D
Solution :
We have,\[f:R\to R,g:R\to R\]defined by \[f(x)=2x-3\]and \[g(x)={{x}^{3}}+5\] It can be checked that\[f(x)\] and\[g(x)\] are bijective functions. \[\therefore \]fog is also bijective and \[(fog)=f(f(x))=f({{x}^{3}}+5)=2({{x}^{3}}+5)-3\] \[=2{{x}^{3}}+7\] \[(fog)(x)=y\Rightarrow 2x{{x}^{3}}+7\Rightarrow x={{\left( \frac{y-7}{2} \right)}^{1/3}}\] \[\therefore \] \[{{(fog)}^{-1}}(x)={{\left( \frac{x-7}{2} \right)}^{1/3}},x\in R\]
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