Question

If $f: R \rightarrow R, g: R \rightarrow R$ are defined by
$f(x)=5 x-3, g(x)=x^{2}+3$, then $g o f^{-1}(3)$ is equal to

• A. $\frac{25}{3}$
• B. $\frac{111}{25}$
• C. $\frac{9}{25}$
• D. $\frac{25}{111}$

Answer 1

Answer: (B)

Given, $f(x)=5\, x-3$
Let $ y=5 \,x-3 $
$ \Rightarrow y+3=5 x $
$ \Rightarrow x=\frac{y+3}{5}$
$\therefore f^{-1}(y)=\frac{y+3}{5}$
$\Rightarrow f^{-1}(x) =\frac{x+3}{5} $
and $g(x) =x^{2}+3 $
Now, $g o f^{-1}(3)=g[f^{-1}(3)]$
$=g(\frac{3+3}{5})=g(\frac{6}{5})$
$\Rightarrow g\left(\frac{6}{5}\right)=\frac{(6)^{2}}{(5)^{2}}+3=\frac{36}{25}+3=\frac{111}{25}$