Invertible function $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $

Question

How can I prove that $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $ is invertible.

First I choose variable $x$ for $y$ and tried to switch and simplified the function but I am stuck. Need some help please.

Answer 1

$$f'(x) = x^2+ 5/3 >0$$

So $f(x)$ is monotonically increasing and hence 1-1. Also the range is the reals so it is invertible.

Answer 2

The function passes the horizontal line test, so it has an inverse function. The fact that the slope is always positive as the user44197 mentioned, implies that it passes the horizontal line test.

Answer 3

Both $g(x)=\frac{x^3}3$ and $h(x)=\frac{5x}3+2$ are increasing functions. (This should be clear if you know graphs of some basic functions.)

Sum of two (strictly) increasing functions is again an increasing function, therefore $f(x)=g(x)+h(x)$ is strictly increasing.

If a function is strictly increasing, then it is injective.