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 where each input corresponds to just one output. That is to say, for each x, there is just one y and vice versa. This implies that the graph of a one-to-one function will never intersect.

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

Let's look at the pictures below:

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For f(x), each value in the left circle corresponds to a unique value in the right circle. Similarly, any value in the right circle correlates to a unique value in the left circle. In mathematical jargon, this implies every domain has a unique range, and every range holds a unique domain. Hence, this is a representation of a one-to-one function.

Here are some different examples of one-to-one functions:

• f(x) = x + 1

• f(x) = 2x

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

Notice that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). For example, the inputs -2 and 2 have identical output, i.e., 4. In the same manner, the inputs -4 and 4 have the same output, i.e., 16. We can see that there are identical Y values for many X values. Hence, 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 characteristics of One to One Functions?

One-to-one functions have these qualities:

• The function has an inverse.

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

• The function passes the horizontal line test.

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

How to Graph a One to One Function

To graph a one-to-one function, you are required to find the domain and range for the function. Let's examine an easy example of a function f(x) = x + 1.

Immediately after you have the domain and the range for the function, you have to plot 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 if a function is one-to-one, we can leverage the horizontal line test. Immediately after you chart 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 spot, 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 deduct all linear functions are one-to-one functions. Keep in mind that we do not apply the vertical line test for one-to-one functions.

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

On the contrary, if the function is not a one-to-one function, it will intersect the same horizontal line more than once. Let's look at the figure 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 crosses various horizontal lines. Case in point, for both domains -1 and 1, the range is 1. Similarly, for either -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 only one input value for each output value, the inverse of a one-to-one function also happens to be a one-to-one function. The opposite of the function essentially reverses the function.

Case in point, in the example of f(x) = x + 1, we add 1 to each value of x in order to get the output, i.e., y. The inverse of this function will deduct 1 from each value of y.

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

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

The qualities of an inverse one-to-one function are no different than every other one-to-one functions. This implies that the inverse of a one-to-one function will possess one domain for every range and pass the horizontal line test.

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

Finding the inverse of a function is simple. You just need to swap the x and y values. For instance, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.

Source

As we reviewed before, the inverse of a one-to-one function reverses the function. Since the original output value required us to add 5 to each input value, the new output value will require us to subtract 5 from each input value.

One to One Function Practice Questions

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 every function:

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

2. Draw the function and its inverse.

3. Determine the inverse of the function mathematically.

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

5. Use the inverse to solve for x in each equation.

Grade Potential Can Help You Master You Functions

If you happen to be facing difficulties using one-to-one functions or similar functions, Grade Potential can put you in contact with a 1:1 teacher who can assist you. Our St Petersburg math tutors are skilled professionals who help students just like you advance their understanding of these subjects.

With Grade Potential, you can learn at your own pace from the comfort of your own home. Plan an appointment with Grade Potential today by calling (727) 605-5897 to find out more about our educational services. One of our consultants will call you to better inquire about your needs to set you up with the best teacher for you!