Question

Let $f$ and $g$ be two differentiable functions satisfying $g\left(a\right)=b$, $g^{\prime }\left(a\right)=2$ and $fog=I$ (where $I$ is the identity function). Then $f^{\prime }\left(b\right)$ is equal to?

• A. $\frac{1}{2}$
• B. $2$
• C. $\frac{2}{3}$
• D. $3$

Answer 1

The correct option is A $\frac{1}{2}$

$f$ and $g$ are two differentiable functions satisfying $g\left(a\right)=b$, $g^{\prime }\left(a\right)=2$ and $fog=I$(identity function).
i.e $fog\left(x\right)=x$ for all x
$\Rightarrow f^{\prime }\left(g\left(x\right)\right)g^{\prime }\left(x\right)=1$ for all x
$\Rightarrow f^{\prime }\left(g\left(a\right)\right)g^{\prime }\left(a\right)=1\therefore f^{\prime }\left(b\right)=\frac{1}{2}$