Proving Inverse Function: f(x)=g-1(x)

[ritwik06]

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.

 

[quasar987]

What you're trying to prove is usually what is taken to be the definition of the inverse function!

What definition are you using then?

 

[ritwik06]

What you're trying to prove is usually what is taken to be the definition of the inverse function!

What definition are you using then?

The definition that I use is that the inverse function is the one which returns the input (as its output) when provided with the output of the original function.

So, Have I proved it right? Are there no exceptions to it?

 

[quasar987]

The definition that I use is that the inverse function is the one which returns the input (as its output) when provided with the output of the original function.

This is not the definition of the inverse (it is the definition of the left inverse). Surely you have the definition of the inverse in mathematical symbols somewhere in your book or class notes?

 

[Tac-Tics]

The definition that I use is that the inverse function is the one which returns the input (as its output) when provided with the output of the original function.

So, Have I proved it right? Are there no exceptions to it?

I agree with quasar that you might want to review your notes or ask your professor for the definition he or she is using.

However, it seems you're not far off the mark. One suitable definition might be that for any functions f and g such that f:A->B, g:B->A, if for each a in A and b in B, f(a) = b and g(b) = a, then g is the inverse of f.