Lesson Objectives

- Demonstrate an understanding of functions
- Learn the definition of a one-to-one function
- Learn how to use the horizontal line test

## How to Determine if a Function is One-to-One Using the Horizontal Line Test

We previously learned that a function is a relation where each x-value corresponds to one and only one y-value. A function is allowed to have a y-value that corresponds to different x-values.

{(3,1),(6,5),(9,1),(2,3)}

In our function above, the y-value of 1 is linked up with two different x-values (3 and 9). This does not violate the definition of a function since each x-value is associated with only one y-value.

If we look at our function from above:

{(3,1),(6,5),(9,1),(2,3)} This function is not a one-to-one function. For each y-value, we do not have one x-value. The y-value of 1 is associated with two different x-values (3 and 9). Let's look at an example.

Example 1: Determine if the function is one-to-one.

{(7,5),(3,8),(2,6),(-7,-1)}

Since each x-value corresponds to one y-value and each y-value corresponds to one x-value, we have a one-to-one function.

Example 2: Use the horizontal line test to determine if the function is one-to-one. $$f(x)=(x - 3)^2 - 1$$ First, let's graph our function. You can sketch this by hand or use an online graphing calculator such as Desmos.com. If we look at our graph, we can see that it fails the horizontal line test. A horizontal line would impact the graph in more than one location, therefore, this is not the graph of a one-to-one function.

Example 3: Use the horizontal line test to determine if the function is one-to-one. $$f(x)=\sqrt{x + 3}- 1$$ First, let's graph our function. You can sketch this by hand or use an online graphing calculator such as Desmos.com. If we look at our graph, we can see that it passes the horizontal line test. No horizontal line would impact the graph in more than one location, therefore, this is the graph of a one-to-one function.

{(3,1),(6,5),(9,1),(2,3)}

In our function above, the y-value of 1 is linked up with two different x-values (3 and 9). This does not violate the definition of a function since each x-value is associated with only one y-value.

### One-to-One Function

A one-to-one function has a stricter definition. For a function to be a one-to-one function, each x-value can correspond to only one y-value and each y-value can correspond to only one x-value.If we look at our function from above:

{(3,1),(6,5),(9,1),(2,3)} This function is not a one-to-one function. For each y-value, we do not have one x-value. The y-value of 1 is associated with two different x-values (3 and 9). Let's look at an example.

Example 1: Determine if the function is one-to-one.

{(7,5),(3,8),(2,6),(-7,-1)}

Since each x-value corresponds to one y-value and each y-value corresponds to one x-value, we have a one-to-one function.

### Horizontal Line Test

When studying functions, we developed the vertical line test to determine if the graph of a relation is a function. For a relation to be a function, no vertical line should intersect the graph in more than one location. If it does, this means an x-value corresponds to more than one y-value, therefore, we do not have a function. Similarly, to determine if we have a one-to-one function, we can use a horizontal line test. The horizontal line test tells us if any horizontal line intersects the graph in more than one location, it is not the graph of a one-to-one function. Since a horizontal line always has the same y-coordinate, if it impacts the graph in more than one location, this tells us the same y-value is associated with more than one x-value, therefore, the function is not a one-to-one function. Let's look at an example.Example 2: Use the horizontal line test to determine if the function is one-to-one. $$f(x)=(x - 3)^2 - 1$$ First, let's graph our function. You can sketch this by hand or use an online graphing calculator such as Desmos.com. If we look at our graph, we can see that it fails the horizontal line test. A horizontal line would impact the graph in more than one location, therefore, this is not the graph of a one-to-one function.

Example 3: Use the horizontal line test to determine if the function is one-to-one. $$f(x)=\sqrt{x + 3}- 1$$ First, let's graph our function. You can sketch this by hand or use an online graphing calculator such as Desmos.com. If we look at our graph, we can see that it passes the horizontal line test. No horizontal line would impact the graph in more than one location, therefore, this is the graph of a one-to-one function.

#### Skills Check:

Example #1

Determine if the Function is one-to-one. $$f(x)=\frac{13}{11}x + 2$$

Please choose the best answer.

A

Yes

B

No

Example #2

Determine if the Function is one-to-one. $$f(x)=\frac{1}{2}(x + 2)^2 - 9$$

Please choose the best answer.

A

Yes

B

No

Example #3

Determine if the Function is one-to-one. $$f(x)=\frac{3}{5}(x - 3)^3 - 1$$

Please choose the best answer.

A

Yes

B

No

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