Lesson Explainer: Absolute Value Functions Mathematics

Example 1: Finding the Range of an Absolute Value Function Using a Graph

Find the range of the function .

Answer

We can see that the graph has a vertex at , a line of symmetry of , an -intercept of , and a -intercept of 2. By inputting values of into the function and observing the resulting outputs, we can confirm the graph’s validity.

Let’s use the graph to help determine the domain and range of . Recall that the domain of a function is the set of all possible input values, while the range is the set of all possible output values. Another way to say this is that the domain is all possible values of the independent variable, while the range is all possible values of the dependent variable.

The arrows tell us that the graph extends infinitely to both the left and the right, so we know that the domain must be the set of all real numbers, or . The arrows also tell us that the graph extends infinitely upward, but we can see that it lies only on or above the -axis. In other words, the smallest value of is 0, but there is no largest value. Thus, we know that the range is , or . Any absolute value function in the form will have this range.

Example 2: Finding the Domain and Range of an Absolute Value Function Using a Graph

Determine the domain and the range of the function .

Answer

We know that the domain and range of remain the same after a horizontal translation to the left or right, so the domain of is the set of all real numbers, or , and the range is , or . However, although the domain of remains the same after a vertical translation up or down, its range does not. The range of is , or .

Here, we can see that the graph is a horizontal translation 1 unit to the left and a vertical translation 1 unit up of the graph of . That is, . Thus, the domain of the function is , and the range is , or .

Note

We will arrive at the same answers for the function’s domain and range by using the fact that the domain is the set of all possible input values, while the range is the set of all possible output values. The arrows tell us that the graph extends infinitely to both the left and the right, so, again, we know that the domain of the function is the set of all real numbers, or . The arrows also tell us that the graph extends infinitely upward, but we can see that the graph lies only at or above a value of 1 for . That is, the smallest value of is 1, but there is no largest value. Thus, we can again see that the range of the function is , or .

Example 3: Finding the Domain of an Absolute Value Function

Given that is a constant, what is the domain of the function ?

Answer

As we can see in the figures below, the graph of is a horizontal translation of the graph of . If is positive, the translation is units to the left, and if is negative, the translation is units to the right.

Recall that for both and , the domain is , and the range is , or . In other words, the domain and range of are the same as those of , regardless of whether the horizontal translation is left or right, and regardless of the number of units. This means the domain of the function must be the set of all real numbers, or .

Note

We know that the domain of a function is the set of all possible input values, or all possible values of the independent variable, . To determine the domain of the function , we could ask ourselves, “What values of can we input into the function?” To help answer this question, let’s consider what the function would be for some different values of the constant .

First we will consider a negative value for . Let’s assume that . When this is the case, the function becomes . Now let’s input some values into the function and observe whether or not an output value is produced:

We can see that all of the input values give a corresponding output value. Now let’s look at what happens when . In this case, the function becomes :

Again, we can see that all of the input values give a corresponding output value. Finally, let’s consider a positive value of a by looking at the function when . Now, the function becomes :

Just as before, there is an output value for each of the input values. In fact, because the absolute value function is defined for all real numbers , no matter what constant represents, we will always be able to input any value into the function and get an output value. That is, the output value will never be undefined. Therefore, we again see that the domain of the function is the set of all real numbers, or .

Example 4: Finding the Domain and Range of an Absolute Value Function

Find the domain and range of the function .

Answer

To help determine the domain and range of the function, let’s begin by constructing its graph. Given , we know that the function’s definition is in the form , where , , and . Thus, we can obtain the graph of by vertically stretching the graph of by a scale factor of 4, reflecting it over the -axis, translating it 5 units to the right, and then translating it 1 unit down as shown:

We can verify that our graph is correct by inputting three values of into the function and observing the resulting output values. This is not necessary as long as we are careful with our transformations, but it is useful for more complicated functions such as this one. The values we choose for will make the value of the expression inside the absolute value symbols , negative, 0, and positive respectively:

We get the ordered pairs , , and . By plotting these points on the coordinate plane, and using our understanding of absolute value functions, we can again create the graph of for the function below:

The arrows on our graph tell us that the graph extends infinitely to both the left and to the right, so we know that the domain of must be the set of all real numbers, or . The arrows also tell us that the graph extends infinitely downward, but we can see that it lies only at or below a value of for . In other words, the largest value of is , but there is no smallest value. This means that the range of the function is , or .

Example 5: Solving a Word Problem Involving an Absolute Value Function

A body was moving with a uniform velocity of magnitude 5 cm/s from the point to the point passing through the point without stopping. The distance between the body and the point is given by , where is the time in seconds, and is the distance in cm. Determine the distance between the body and the point after 5 seconds and after 11 seconds.

Answer

In order to determine the distance between the body and the point after 5 seconds and after 11 seconds, we need to evaluate the function at and at . In other words, we must substitute both 5 and 11 into the function for and calculate the resulting value of .

Recall that the absolute value of a number is always positive, so regardless of whether the value of the expression inside the absolute value symbols is positive or negative, its absolute value will be positive. This ensures that the distance given by the function will be positive, because the product of 5 and another positive number will always be positive. First, let’s find :

Based on our calculations, we know that, after 5 seconds, the distance between the body and point was 15 centimetres.

Now let’s find :

Our calculations show us that the distance between the body and point was also 15 centimetres after 11 seconds. Notice that our two distances are the same. This tells us that after 5 seconds, the body must not have yet reached point and that after 11 seconds, it must have already passed point .