﻿ One to One Function is the inverse of a function. A 1-to-1 function is just...

# One to One Function

Range, Domain, horizontal line test

#### What is a 1 to 1 function?

A special type of function.

Look at the two functions below, (# 1 and #2). They only differ by a single number .

One of the functions is one to one , and the other is not.

##### Which function below is 1 to 1?

Function #2 on the right side is the one to one function . In a one to one function, every element in the range corresponds with one and only one element in the domain.

So, #1 is not one to one because the range element. 5 goes with 2 different values in the domain (4 and 11).

Diagram 1

#### How to know if $$f(x)$$ is 1 to 1?

To be a function

Only one thing must be true : each element in domain must go to a unique range element. ( prior lesson)

Diagram 2

To be a 1 to 1 function

Two things must be true.

First: It must be a standard function. In other words, it must satisfy requirements for function .

Second: This is the new part. each element in range must go to a unique element in the domain.

Diagram 3
So, there is one new characteristic that must be true for a function to be one to one. This new requirement can also be seen graphically when we plot functions, something we will look at below with the horizontal line test.

### Horizontal Line Test

We can also look at the graphs of functions and use the horizontal line test to determine whether or not a function is one to one.

To be a function

Only one thing must be true : The relation must pass the vertical line test. ( link) .

As we learned in our vertical line test lesson, this is really the exact same as saying "elements in the domain cannot repeat".

To be a 1 to 1 function

Two things must be true.

First: It must be a standard function. In other words, it must pass the vertical line test.

Second: This is the new part. It must also pass the horizontal line Test .

### Arrow Charts

Arrow Chart of 1 to 1 vs Regular Function

Below you can see an arrow chart diagram that illustrates the difference between a regular function and a one to one function.

Function #1 is not a 1 to 1 because the range element of '5' goes with two different elements (4 and 11) in the domain. .

### Practice Problems - Part I

##### Problem 1

Which functions below are 1 to 1?

• Function #1 {(2, 27), (3, 28), (4, 29), (5, 30)}
• Function #2 {(11,14), (12, 14), (16, 7), (18, 13)}
• Function #3 {(3, 12), (4, 13), (6, 14), (8, 1)}

If an element in the range repeats, like 14 in function #2 , then you do not have a 1 to 1 function.

Relation #1 and Relation #3 are both one-to-one functions.

##### Problem 2

Which functions below are one to one?

• Function #1 {(2, 1), (4, 5), (6, 7), (8, 9)}
• Function #2 {(3,6), (8, 5), (6, 7), (22,6)}
• Function #3 {(-3, 4), (21, -5), (0, 0), (8, 9)}
• Function #4 {(9, 19), (34, 5), (6, 17), (8, 19)}

If an element in the range repeats, like 6 in function #2 or 19 in function #4, then you do not have a 1 to 1 function.

Relation #1 and Relation #3 are both one-to-one functions.

##### Problem 3

Yes, because every element in the range is matched with only 1 element in the domain.

### Practice Problems - Part II

Use the horizontal line test and your knowledge of 1 to 1 functions to determine whether or not each graph below is 1 to 1.

##### Problem 1

Ask yourself: "Can I draw a horizontal line (anywhere) that will hit the graph two times?"

Since the answer is 'yes', this is not a one-to-one function.

##### Problem 2

Ask yourself: "Can I draw a horizontal line (anywhere) that will hit the graph two times?"

Since the answer is 'no', this is a one-to-one function.

##### Problem 3

Ask yourself: "Can I draw a horizontal line (anywhere) that will hit the graph two times?"

Since the answer is 'no', this is a 1 to 1 function.

##### Problem 4

Vertical lines such as x = 2 are not functions at all.

Horizontal linessuch as y = 9 are functions but they are not 1 to 1 functions.

All other lines are indeed one to one functions.

##### Problem 5
No

As the picture below shows, parabolas are not one to one.