Proove that f(x)=x^3+x has inverse function

For the given function to have an inverse function
f^-1(x), it has to be bijective.

For the function to be
bijective, it has to be one to one function and on-to
function.

We'll prove that the given function is one to one
function.

For this reason, we'll have to prove that the
function is strictly increasing or decreasing. For this reason, we'll determine the
first derivative of the function.

f'(x) =
(x^3+x)'

f'(x) = 3x^2 + 1

It
is obvious that f'(x)>0 for any value of x, so the function f(x) is an one to one
function.

Let's prove that the function is an on to
function. We'll determine the limits of the function for x-> +infinite and
x->-infinite

Lim f(x) = lim (x^3 + x) =
(+infinite)^3 + infinite = +infinite

Lim f(x) = lim (x^3 +
x) = (-infinite)^3 - infinite = -infinite

So, f(x) is a
continuous function, then it is an on-to
function.

Since the fuction is both, one to
one and on to function, it is bijective and it does exist
f^-1(x).