Graphing the Domain and Range of Functions | Math

August 11, 2015

Recently, you started building your understanding of functions. (See, they’re not that bad.) Functions relate two sets of numbers called the domain and the range. The domain of a function is the set of all first elements in an ordered pair (the x-coordinates). The range is the set of all second elements in an ordered pair (the y-coordinates). Incorporating the domain and range of a graph or given formula into your understanding of functions is a high emphasis topic on the TASC Test Assessing Secondary Completion™ Mathematics subtest.

Let’s dive right into some examples from the Khan Academy:

Domain and Range from a Graph

Example 1: Consider the function h in the graph:

• What is the domain of the function?

_____ ≤ t ≤ _____

• What is the range of the function?

_____ ≤ h(t) ≤ _____

Example 2: Consider the function f in the graph:

• What is the domain of the function?

_____ ≤ t ≤ _____

• What is the range of the function?

_____ ≤ f(t) ≤ _____

Answers: To find the domain, follow the x axis (shown as the t axis in these examples). Start with the negative quadrants first, since the domain is “greater than or equal to” (≤) that number. The same is true for finding the range. Follow the the y axis (shown as the f(t) or h(t) axis in these examples). Start with the negative quadrants first, since the range is “greater than or equal to” (≤) that number.

Example 1:

What is the domain of the function?

-4 ≤ t ≤ 8

What is the range of the function?

-8 ≤ h(t) ≤ 8

Example 2:

What is the domain of the function?

-6 ≤ t ≤ 6

What is the range of the function?

-8 ≤ f(t) ≤ 9

Domain and Range from a Formula

Determine the domain and range of the function f(x) = 3x2 + 6x -2.

To find the domain of the function, develop a set of all valid inputs (x-values) of the function. You could essentially enter any real number in for the x-value.

Domain = all real numbers.

To find the range of the function, solve for the set of possible outputs (y-values) of the function. Remember, y= f(x). Create your x and y coordinates (since the domain = all real numbers):

 x coordinates y coordinate -2 3(-22) – 6(-2) - 2 =-2 -1 3(-12) – 6(-1) – 2 = -5 0 3(02) – 6(0) – 2 = -2 1 3(12) – 6(1) – 2 = 7

Plot these coordinates to find the range:

The vertex of the graphed parabola is (-1,-5) because it’s the lowest point on the graph. The range can now be found with this information. Range = all real numbers ≥ -5.

Continue practicing with finding the domain and range of a function for the TASC test at the Khan Academy.