If $f: R \rightarrow R, g: R \rightarrow \mathrm{R}$ are defined by $f(x)=5 x-3, g(x)$ $=x^2+3$, then $g…
If $f: R \rightarrow R, g: R \rightarrow \mathrm{R}$ are defined by $f(x)=5 x-3, g(x)$ $=x^2+3$, then $g \circ f^{-1}(3)$ is equal to
- $\frac{25}{3}$
- $\frac{111}{25}$
- $\frac{9}{25}$
- $\frac{25}{111}$
Solution
Given, $f(x)=5 x-3$ and $g(x)=x^2+3$
Let, $y=f(x), \therefore y=5 x-3$
$
\begin{aligned}
& 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}
\end{aligned}
$
Now, $\quad g(x)=x^2+3$;
So, $g \circ f^{-1}(3)=g\left[f^{-1}(3)\right]$
$
=g\left(\frac{3+3}{5}\right)=g\left(\frac{6}{5}\right)=\frac{(6)^2}{(5)^2}+3=\frac{36}{25}+3=\frac{111}{25}
$
Asked in: AP EAMCET 2015
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