Question

Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and f' (3) = 9, write the value of g' (9).

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},f\left(3\right)=9,f\text{'}\left(3\right)=9 \\ \mathrm{and}g\left(x\right)=f^{-1}\left(x\right) \\ \Rightarrow \left(gof\right)\left(x\right)=x \\ \Rightarrow g\left\{f\left(x\right)\right\}=x \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{d}{dx}\left[g\left\{f\left(x\right)\right\}\right]=1 \\ \Rightarrow g\text{'}\left\{f\left(x\right)\right\}\frac{d}{dx}\left\{f\left(x\right)\right\}=1 \\ \Rightarrow g\text{'}\left\{f\left(x\right)\right\}\times f\text{'}\left(x\right)=1 \\ \mathrm{Puting}x=3,\mathrm{we}\mathrm{get}, \\ g\text{'}\left\{f\left(3\right)\right\}\times f\text{'}\left(3\right)=1 \\ \Rightarrow g\text{'}\left(9\right)\times 9=1\left[\because f\left(3\right)=9,f\text{'}\left(3\right)=9\right] \\ \Rightarrow g\text{'}\left(9\right)=\frac{1}{9} \end{gathered}$$