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.

Possible Answers:

True

False

Correct answer:

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 ?

Possible Answers:

Correct answer:

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?

Possible Answers:

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.

Correct answer:

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  ?

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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.

Possible Answers:

False

True

Correct answer:

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 ?

Possible Answers:

Correct answer:

Explanation:

First, we must consider the graph of .

Using the formula  we can apply the transformations step-by-step.  First we will transform the amplitude, so  so we must shorten the amplitude to .

Now we must apply a vertical shift of one unit since .  This leaves us with our answer.