$\"\"$

\

The function is $\"\"$ .

\

Horizontal line test:

\

A function $\"\"$ has an inverse $\"\"$ if and only if no horizontal line intersects its graph of the function at most one point.

\

Graph :

\

Graph the function $\"\"$ .

\

Draw the horizontal line $\"\"$ .

\

$\"\"$

\

Observe the graph :

\

The horizontal line $\"\"$ touches the graph of the function at only one point.

\

The function is an one-to-one, because it passes the horizontal line test.

\

Therefore, the inverse of the function exist.

\

$\"\"$

\

$\"\"$ .

\

To find the inverse of the function, consider $\"\"$ and solve $\"\"$ in terms of $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Squaring on each side.

\

$\"\"$

\

Now interchange $\"\"$ and $\"\"$ .

\

$\"\"$

\

Replace $\"\"$ .

\

$\"\"$ .

\

The inverse of $\"\"$ is $\"\"$ .

\

$\"\"$

\

Find the domain of $\"\"$ .

\

For the rational function, the denominator of the function should not be zero and the function under the square root should not be zero.

\

$\"\"$

\

Domain of the function $\"\"$ is $\"\"$ .

\

Range of the function $\"\"$ is $\"\"$ .

\

Find the domain of $\"\"$ .

\

For the rational function, the denominator of the function should not be zero.

\

$\"\"$

\

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

\

Restrictions over the domain of $\"\"$ :

\

The domain of inverse funtion $\"\"$ is the range of the function $\"\"$ .

\

Range of the function $\"\"$ is $\"\"$ .

\

Hence the domain of $\"\"$ is $\"\"$ .

\

$\"\"$

\

Therefore the inverse of the function $\"\"$ .

\

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