$\"\"$

The function is $\"\"$ .

Since the function is polynomial, the domain of the function is all real numbers.

The domain of $\"\"$ is $\"\"$ .

$\"\"$

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

Apply power rule in derivatives $\"\"$ .

$\"\"$ .

$\"\"$ on $\"\"$ .

$\"\"$ at $\"\"$ .

The derivative is always positive on $\"\"$ therefore $\"\"$ is increasing and thus has an inverse on this interval.

$\"\"$ is positive on the domain of $\"\"$ .

$\"\"$ is strictly monotonic and it must have inverse function.

Inverse exists.

$\"\"$

Inverse exists.