- #1

ritwik06

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## Homework Statement

The problem is to prove that:

if

f(g(x))=x ... (1)

and

g(f(x))=x ...(2)

then f(x)=g

^{-1}(x)

## The Attempt at a Solution

Differentiating (1) wrt x

f'(g(x))*g'(x)=1

f'(g(x))=1/g'(x)

As the slopes are reciprocals of each other, hence f(x)=g

^{-1}(x)

Is this as simple as it seems? Are there any possible exceptions?

Is there any more explanatory proof?

Please help me.