Question

Calculus - inverse functions:
If f(x) = x^5 + x^3 + x, find f((f^-1)(2)). Can anyone please teach me how to solve this?

Answer 1

Ok, so by definition if g is the inverse function of f (and some technical stuff about domains of functions), then
g(f(x)) = x for all x for which this makes sense
and f(g(x)) = x , for all x for which this makes sense

So in your case ...

Answer 2

f'(x) = 5x^4 + 3x^2 + 1 > 0 for all x if and only if an inverse of f(x) exists.

Since powers are even, f'(x) > 0 for all x, guaranteed.

Then since the inverse exists, f((f^-1)) exists. Thus if f(x) = a, then f(a)^(-1) = x (i.e. the inverse of f evaluated at a) and f(x) = f(f(a)^(-1) = a.

In our case, a = 2

Ans. f((f^-1)(2)) = 2