A) \[{{\left( \frac{x-7}{3} \right)}^{1/2}}\]
B) \[{{\left( \frac{x+7}{3} \right)}^{1/2}}\]
C) \[{{\left( \frac{x-3}{7} \right)}^{1/2}}\]
D) \[{{\left( \frac{x+3}{7} \right)}^{1/2}}\]
Correct Answer: A
Solution :
Given \[f(x)=3x+10 \,and g(x)={{x}^{2}}-1\] \[\Rightarrow \,\,\,fog=\,\,f(g(x))=\,\,f({{x}^{2}}-1)\] \[=\,\,\,3\left( {{x}^{2}}-1 \right)+10=3{{x}^{2}}+7\] ... (i) Let \[3{{x}^{2}}+7=y\,\,\Rightarrow \,\,\,3{{x}^{2}}=y-7\] \[\Rightarrow \,\,\,{{x}^{2}}=\frac{y-7}{3}\,\,\Rightarrow \,\,x={{\left( \frac{y-7}{3} \right)}^{1/2}}\] \[\because \,\,\,f(x)=y,\,\,\,\Rightarrow \,\,\,x=\,\,{{f}^{-1}}\,(y)\] \[\therefore \,\,{{(fog)}^{-1}}={{\left( \frac{x-7}{3} \right)}^{1/2}}\]
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