### All Trigonometry Resources

## 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

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:**

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.

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

Which of the following represents the graph of with ?

**Possible Answers:**

**Correct answer:**

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:**

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:**

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

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

Which of the following is the graph of ?

**Possible Answers:**

**Correct answer:**

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.

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