What is the inverse of $f(x) = x^3/6 - 36/x$?

Question

The question is:

The function $f$ is defined by:

$$f(x) = \frac{1}{6}x^3 - \frac{36}{x} , x > 0$$

Show that the function $f$ has an inverse.

I tried to use an inverse function calculator online but the result is extremely complicated.

Answer 1

Alternatively you could plot the function $f(x) = x^3/6 - 36/x$ for $x>0$ and apply the horizontal line test.

If any horizontal line $y=c$ is drawn on the above graph and intersects the graph of $f(x)$ more than once. The function is not injective i.e. we have the same $y$ value for two different values of $x$ and hence $f(x)$ is not one-to-one (injective). To add more lingo - the particular function you supplied is bijective because the horizontal line will only intercept $f(x)$ exactly once. This is a necessary condition for a function to have an inverse.

Answer 2

$f'(x)=\frac{1}{2}x^2+\frac{36}{x^2}>0$ for any $x>0$ so this is an increasing function .

Therefore $f(x)$ has an inverse.