Q53 / 100 · BITSAT 2024 (Memory Based Paper 2) · Mathematics

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

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

Solution

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

Hence, the correct answer is option: 'B'.

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