Q231 / 270 · NDA 2015 (Phase 2) · Mathematics

If $f: R \rightarrow R, g: R \rightarrow R$ be two functions given by $f(x)=2 x-3$ and $g(x)=x^3+5$, then $(f \circ g)^{-1}(x)$ is equal to

  • A
    $\left(\frac{x+7}{2}\right)^{\frac{1}{3}}$
  • Correct answer
    B
    $\left(\frac{x-7}{2}\right)^{\frac{1}{3}}$
  • C
    $\left(x-\frac{7}{2}\right)^{\frac{1}{3}}$
  • D
    $\left(x+\frac{7}{2}\right)^{\frac{1}{3}}$

Solution

$\begin{aligned} & f(x)=2 x-3 \\ & g(x)=x^3+5 \\ & \operatorname{fog}(x)=f[g(x)]=f\left(x^3+5\right) \\ & =2\left(x^3+5\right)-3 \\ & \operatorname{fog}(x)=2 x^3+7=y(\text { say })\end{aligned}$ $\begin{aligned} & \Rightarrow 2 x^3=y-7 \\ & \Rightarrow \quad x=\left[\frac{y-7}{2}\right]^{\frac{1}{3}} \\ & \Rightarrow \quad \operatorname{fog}^{-1}(y)=\left[\frac{y-7}{2}\right]^{\frac{1}{3}} \\ & \Rightarrow \quad \operatorname{fog}^{-1}(x)=\left[\frac{x-7}{2}\right]^{\frac{1}{3}}\end{aligned}$

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

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