Try refreshing the page, or contact customer support.

You must create an account to continue watching

Register to view this lesson

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 79,000
lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you
succeed.

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn what makes a one-to-one function different from a regular function. Learn a simple test you can use to check whether a function is one-to-one or not.

A Function

A good way of describing a function is to say that it gives you an output for a given input. You give functions a certain value to begin with and they do their thing on the value, and then they give you the answer. For example, the function f(x) = x + 1 adds 1 to any value you feed it. You give it a 5, this function will give you a 6: f(5) = 5 + 1 = 6.

Functions do have a criterion they have to meet, though. And that is the x value, or the input, cannot be linked to more than one output or answer. In other words, you cannot feed the function one value and end up with two different answers. For example, if you give a supposed function a 1 and it gives you a 4 and a 10, then you know that this supposed function is not a real function. A real function would give you one solid answer only.

A One-to-One Function

Now, let's talk about one-to-one functions. A one-to-one function is a function in which the answers never repeat. A normal function can have two different input values that produce the same answer, but a one-to-one function does not. For example, the function f(x) = x^2 is not a one-to-one function because it produces 4 as the answer when you input both a 2 and a -2, but the function f(x) = x - 3 is a one-to-one function because it produces a different answer for every input.

Graphs

We can learn a lot by comparing graphs of functions that are and are not one-to-one functions. Let's look at one that is and one that is not a one-to-one function.

The function shown here is f(x) = x + 2, and it is a one-to-one function.

This is a one-to-one function.

You can see that every input, x, produces a different answer, y.

Now, let's look at the graph of f(x) = x^2, which is not a one-to-one function.

This is not a one-to-one function.

This function is not a one-to-one function because we have two different input values, x, that produce the same answer, y. Look at the graph when the input is both a 3 and a -3. You can see that both produce 9 as the answer.

Unlock Content

Over 79,000 lessons in all major subjects

Get access risk-free for 30 days,
just create an account.

By comparing these two graphs, we can see that the horizontal line test works very well as an easy test to see if a function is one-to-one or not. The horizontal line test tells us that if you draw a line and the graph crosses the horizontal more than once, then the function is not a one-to-one function.

Looking at our second graph of f(x) = x^2, we see that if we draw a horizontal line, our graph crosses that line twice, which is more than once. Because our graph crosses the horizontal line more than once, we see that this function is not a one-to-one function.

Lesson Summary

What have we learned? We've learned that a function gives you an output for a given input. A one-to-one function is a function of which the answers never repeat. For example, the function f(x) = x + 1 is a one-to-one function because it produces a different answer for every input. The function f(x) = x^2, on the other hand, is not a one-to-one function because it gives you the same answer for more than one input. This particular function gives you 9 when you give it either a 3 or a -3. A one-to-one function would not give you the same answer for both inputs.

An easy way to test whether a function is one-to-one or not is to apply the horizontal line test to its graph. If the graph crosses the horizontal line more than once, then the function is not a one-to-one function.

Learning Outcomes

As you go through this lesson, you can prepare to:

Understand what constitutes a function

Compare graphs

Contrast functions and one-to-one functions

Use the horizontal line test to determine whether a function is a one-to-one function

Did you know… We have over 200 college
courses that prepare you to earn
credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the
first two years of college and save thousands off your degree. Anyone can earn
credit-by-exam regardless of age or education level.

Not sure what college you want to attend yet? Study.com has thousands of articles about every
imaginable degree, area of
study
and career path that can help you find the school that's right for you.

Research Schools, Degrees & Careers

Get the unbiased info you need to find the right school.