Differentiability

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Question

$Let f and g be differentiable functions satisfying g (a) = b, g^{1} (a) = 2 and f o g = I (identity$ $function). Then f^{1} (b) is equal to$

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Solution

$f o g = I \Rightarrow f o g (x) = x \forall x$
$\begin{array}{l} \Rightarrow f^{1} (g (x)) g^{1} (x) = 1 \forall x \\ \Rightarrow f^{1} (g (a)) = \frac{1}{g^{1} (a)} = \frac{1}{2} \\ \Rightarrow f^{1} (b) = \frac{1}{2} (∵ g (a) = b) \end{array}$

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