May 27, 2022

One to One Functions - Graph, Examples | Horizontal Line Test

What is a One to One Function?

A one-to-one function is a mathematical function whereby each input correlates to just one output. So, for every x, there is only one y and vice versa. This means that the graph of a one-to-one function will never intersect.

The input value in a one-to-one function is noted as the domain of the function, and the output value is the range of the function.

Let's examine the pictures below:

One to One Function


For f(x), every value in the left circle corresponds to a unique value in the right circle. Similarly, each value on the right correlates to a unique value on the left. In mathematical words, this signifies every domain has a unique range, and every range has a unique domain. Hence, this is an example of a one-to-one function.

Here are some additional representations of one-to-one functions:

  • f(x) = x + 1

  • f(x) = 2x

Now let's examine the second picture, which shows the values for g(x).

Pay attention to the fact that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). Case in point, the inputs -2 and 2 have identical output, i.e., 4. In conjunction, the inputs -4 and 4 have identical output, i.e., 16. We can see that there are identical Y values for multiple X values. Therefore, this is not a one-to-one function.

Here are additional representations of non one-to-one functions:

  • f(x) = x^2

  • f(x)=(x+2)^2

What are the qualities of One to One Functions?

One-to-one functions have these qualities:

  • The function owns an inverse.

  • The graph of the function is a line that does not intersect itself.

  • It passes the horizontal line test.

  • The graph of a function and its inverse are equivalent regarding the line y = x.

How to Graph a One to One Function

When trying to graph a one-to-one function, you are required to find the domain and range for the function. Let's look at a simple representation of a function f(x) = x + 1.

Domain Range

Immediately after you possess the domain and the range for the function, you need to graph the domain values on the X-axis and range values on the Y-axis.

How can you determine whether a Function is One to One?

To test whether or not a function is one-to-one, we can leverage the horizontal line test. Once you plot the graph of a function, trace horizontal lines over the graph. In the event that a horizontal line intersects the graph of the function at more than one place, then the function is not one-to-one.

Because the graph of every linear function is a straight line, and a horizontal line will not intersect the graph at more than one point, we can also conclude all linear functions are one-to-one functions. Remember that we do not apply the vertical line test for one-to-one functions.

Let's study the graph for f(x) = x + 1. Once you chart the values for the x-coordinates and y-coordinates, you have to review whether or not a horizontal line intersects the graph at more than one place. In this example, the graph does not intersect any horizontal line more than once. This signifies that the function is a one-to-one function.

Subsequently, if the function is not a one-to-one function, it will intersect the same horizontal line multiple times. Let's examine the graph for the f(y) = y^2. Here are the domain and the range values for the function:

Here is the graph for the function:

In this case, the graph meets numerous horizontal lines. For instance, for either domains -1 and 1, the range is 1. Similarly, for both -2 and 2, the range is 4. This implies that f(x) = x^2 is not a one-to-one function.

What is the inverse of a One-to-One Function?

Considering the fact that a one-to-one function has just one input value for each output value, the inverse of a one-to-one function is also a one-to-one function. The inverse of the function basically undoes the function.

For Instance, in the case of f(x) = x + 1, we add 1 to each value of x as a means of getting the output, or y. The inverse of this function will subtract 1 from each value of y.

The inverse of the function is denoted as f−1.

What are the characteristics of the inverse of a One to One Function?

The characteristics of an inverse one-to-one function are identical to all other one-to-one functions. This means that the inverse of a one-to-one function will hold one domain for each range and pass the horizontal line test.

How do you determine the inverse of a One-to-One Function?

Figuring out the inverse of a function is simple. You simply have to change the x and y values. For example, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.


As we discussed earlier, the inverse of a one-to-one function undoes the function. Since the original output value required us to add 5 to each input value, the new output value will require us to deduct 5 from each input value.

One to One Function Practice Examples

Consider these functions:

  • f(x) = x + 1

  • f(x) = 2x

  • f(x) = x2

  • f(x) = 3x - 2

  • f(x) = |x|

  • g(x) = 2x + 1

  • h(x) = x/2 - 1

  • j(x) = √x

  • k(x) = (x + 2)/(x - 2)

  • l(x) = 3√x

  • m(x) = 5 - x

For any of these functions:

1. Figure out whether the function is one-to-one.

2. Draw the function and its inverse.

3. Figure out the inverse of the function numerically.

4. Specify the domain and range of every function and its inverse.

5. Apply the inverse to find the solution for x in each equation.

Grade Potential Can Help You Learn You Functions

If you are struggling trying to learn one-to-one functions or similar functions, Grade Potential can put you in contact with a private tutor who can support you. Our Mountain View math tutors are skilled professionals who help students just like you enhance their skills of these subjects.

With Grade Potential, you can study at your unique pace from the convenience of your own home. Plan a call with Grade Potential today by calling (650) 459-5193 to find out more about our educational services. One of our representatives will call you to better determine your requirements to find the best instructor for you!

Let Grade Potential set you up with the perfect Grammar tutor!

Or answer a few questions below to get started