Lesson Objectives

- Demonstrate an understanding of functions
- Learn how to determine if a function is one-to-one

## 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 the 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 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.

{(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 two different x-values, 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 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.

#### 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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