$\"\"$

The function is $\"\"$ and the interval is $\"\"$ .

Consider $\"\"$ .

Differentiate on each side with respect to $\"\"$ .

$\"\"$

Apply property : $\"\"$ .

$\"\"$

Since the function $\"\"$ is a piecewise function, it is undefined at $\"\"$ .

$\"\"$

$\"\"$ 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 an inverse function.

$\"\"$

$\"\"$ is monotonically increasing on the interval $\"\"$ , so it must have an inverse function on the interval.