# Trigonometry : Graphs of Inverse Trigonometric Functions

## Example Questions

### Example Question #1 : Graphs Of Inverse Trigonometric Functions

True or False: The inverse of the function  is also a function.

True

False

False

Explanation:

Consider the graph of the function .  It passes the vertical line test, that is if a vertical line is drawn anywhere on the graph it only passes through a single point of the function. This means that  is a function.

Now, for its inverse to also be a function it must pass the horizontal line test. This means that if a horizontal line is drawn anywhere on the graph it will only pass through one point.

This is not true, and we can also see that if we graph the inverse of  () that this does not pass the vertical line test and therefore is not a function.  If you wish to graph the inverse of , then you must restrict the domain so that your graph will pass the vertical line test.

### Example Question #1 : Graphs Of Inverse Trigonometric Functions

Which of the following is the graph of the inverse of  with ?

Explanation:

Note that the inverse of  is not , that is the reciprocal.  The inverse of  is  also written as .  The graph of  with  is as follows.

And so the inverse of this graph must be the following with  and

### Example Question #3 : Graphs Of Inverse Trigonometric Functions

Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?

The  and  values are switched so where  for the parent function,  for the inverse function.

Let  so where  for the parent function,  for the inverse function.

The  values are swapped with  so where  for the parent function,  for the inverse function.

Let  so where  for the parent function  for the inverse function.

The  and  values are switched so where  for the parent function,  for the inverse function.

Explanation:

To find an inverse function you swap the and values.  Take  for example, to find the inverse we use the following method.

(swap the  and  values)

(solving for )

### Example Question #2 : Graphs Of Inverse Trigonometric Functions

Which of the following represents the graph of  with  ?

Explanation:

If we are looking for the graph of  with , that means this is the inverse of   with .  The graph of  with  is

Switching the  and   values to graph the inverse we get the graph

### Example Question #3 : Graphs Of Inverse Trigonometric Functions

Which of the following is the graph of  with ?

Explanation:

We first need to think about the graph of the function .

Using the formula  where  is the vertical shift, we have to perform a transformation of moving the function  up two units on the graph.

### Example Question #6 : Graphs Of Inverse Trigonometric Functions

Which of the following is the correct graph and range of the inverse function of   with ?

Explanation:

First, we must solve for the inverse of

So now we are trying to find the range of and plot the function .  Let’s start with the graph of .  We know the domain is .

Now using the formula  where  = Period, the period of   is .  And so we perform a transformation to the graph of  to change the period from  to .

We can see that the graph has a range of

### Example Question #7 : Graphs Of Inverse Trigonometric Functions

True or False: The domain for  will always be all real numbers no matter the value of  or any transformations applied to the tangent function.

False

True

True

Explanation:

This is true because just as the range of  is all real numbers due to the vertical asymptotes of the function, the function  extends to all values of  but is limited in its values of  .  No matter the transformations applied, all values of will still be reached.

### Example Question #8 : Graphs Of Inverse Trigonometric Functions

Which of the following is the graph of ?