The inverse function  of a function  can be found as follows:

Replace  with :

Switch the positions of  and :

,

or,

Solve for  - that is, isolate it on one side.

Take the reciprocals of both sides:

Multiply both sides by 5:

Add 7:

The right expression can be simplified as follows:

Replace  with :

The inverse function  of a function  can be found as follows:

Replace  with :

Switch the positions of  and :

, 

or

Take the natural logarithm of both sides:

By definition, , so

Add 3 to both sides:

Replace  with :

This is not given among the choices; however, remember that by one of the properties of logarithms,

,

so 

By another property, , so

or

,

which is among the choices and is the correct answer.

A function  has an inverse function if and only if, for all  in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to not be the case:

 and . Since two range values exist to which more than one domain value is matched, the function has no inverse.

The inverse function  of a function  can be found as follows:

Replace  with :

Switch the positions of  and :

, or

Solve for  - that is, isolate it on one side.

First, subtract 15:

Multiply by :

Distribute:

Replace  with :

,

the correct response.

The inverse function  of a function  can be found as follows:

Replace  with :

Switch the positions of  and :

,

or

Solve for . This can be done as follows:

Square both sides:

Add 5 to both sides:

Multiply both sides by , distributing on the right:

Replace  with :

,

the correct response.

Given the graph of , the graph of its inverse,  is the reflection of the former about the line . This line is in dark green below; critical points are reflected as shown:

The figure in blue is the graph of 

A relation is a function if and only if it passes the Vertical Line Test (VLT) - that is, no vertical line exists that passes through its graph more than once. From the diagram below, we see that no such line exists:

The relation passes the VLT, so it is a function.

A function has an inverse if and only if it passes the Horizontal Line Test (HLT) - that is, no horizontal line exists that passes through its graph more than once. From the diagram below, we see that no such line exists:

The function passes the HLT, so it has an inverse.

The inverse function  of a function  can be found as follows:

Replace  with :

Switch the positions of  and :

or

Solve for  - that is, isolate it on one side - as follows:

Raise  to the power of both sides:

A property of logarithms states that , so

Subtract 7 from both sides:

Multiply both sides by , distributing on the right:

Replace  with :

A relation is a function if and only if it passes the Vertical Line Test (VLT)—that is, no vertical line exists that passes through its graph more than once. From the diagram below, we see that no such line exists:

A function has an inverse, if and only if, it passes the Horizontal Line Test (HLT) - that is, no horizontal line exists that passes through its graph more than once. From the diagram below, we see at least one such line exists.

One definition of the inverse function  of a function  is the set of all ordered pairs  such that the ordered pair  is in the set of ordered pairs in . As such, if the ordered pairs of  are given, as is the case here, the set of ordered pairs in  can be found by switching the positions of the coordinates in all of the pairs. Doing this, we obtain:

or, ordering the -coordinates,