Graphs of Inverse Trigonometric Functions - Trigonometry
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True or False: The inverse of the function 
is also a function.
True or False: The inverse of the function 
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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.
Consider the graph of the 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 
Which of the following is the graph of the inverse of 
with 
?
Which of the following is the graph of the inverse of 
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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 
Note that the inverse of 
And so the inverse of this graph must be the following with 
Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?
Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?
Tap to see back →
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 
)
To find an inverse function you swap the 
Which of the following represents the graph of 
with 
?
Which of the following represents the graph of 
Tap to see back →
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
If we are looking for the graph of 
Switching the 
Which of the following is the graph of 
with 
?
Which of the following is the graph of 
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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.
We first need to think about the graph of the function 
Using the formula 
Which of the following is the correct graph and range of the inverse function of 
with 
?
Which of the following is the correct graph and range of the inverse function of 
Tap to see back →
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 
First, we must solve for the inverse of
So now we are trying to find the range of and plot the function 
Now using the formula 
We can see that the graph has a range of
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.
True or False: The domain for 
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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.
This is true because just as the range of 
Which of the following is the graph of 
?
Which of the following is the graph of 
Tap to see back →
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.
First, we must consider the graph of 
Using the formula 
Now we must apply a vertical shift of one unit since 
True or False: The inverse of the function is also a function.
True or False: The inverse of the function
Tap to see back →
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.
Consider the graph of the 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
Which of the following is the graph of the inverse of with ?
Which of the following is the graph of the inverse of
Tap to see back →
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
Note that the inverse of
And so the inverse of this graph must be the following with
Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?
Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?
Tap to see back →
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 )
To find an inverse function you swap the
Which of the following represents the graph of with ?
Which of the following represents the graph of
Tap to see back →
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
If we are looking for the graph of
Switching the
Which of the following is the graph of with ?
Which of the following is the graph of
Tap to see back →
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.
We first need to think about the graph of the function
Using the formula
Which of the following is the correct graph and range of the inverse function of with ?
Which of the following is the correct graph and range of the inverse function of
Tap to see back →
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
First, we must solve for the inverse of
So now we are trying to find the range of and plot the function
Now using the formula
We can see that the graph has a range of
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.
True or False: The domain for
Tap to see back →
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.
This is true because just as the range of
Which of the following is the graph of ?
Which of the following is the graph of
Tap to see back →
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.
First, we must consider the graph of
Using the formula
Now we must apply a vertical shift of one unit since
True or False: The inverse of the function is also a function.
True or False: The inverse of the function
Tap to see back →
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.
Consider the graph of the 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
Which of the following is the graph of the inverse of with ?
Which of the following is the graph of the inverse of
Tap to see back →
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
Note that the inverse of
And so the inverse of this graph must be the following with
Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?
Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?
Tap to see back →
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 )
To find an inverse function you swap the
Which of the following represents the graph of with ?
Which of the following represents the graph of
Tap to see back →
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
If we are looking for the graph of
Switching the