$\"\"$

The function is $\"\"$ .

Graph the function $\"\"$ .

$\"\"$

Observe the graph, the function is decreasing over its domain.

To verify this find the derivative of the $\"\"$ .

$\"\"$ .

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

$\"\"$

Above derivative function is negative for all real values of $\"\"$ .

$\"\"$ is decreasing function

Therefore, $\"\"$ is strictly monotonic and has an inverse function.

$\"\"$

Inverse function exists because the function $\"\"$ is monotonic.