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) = 3x^{2} + 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(-2^{2}) – 6(-2) - 2 =-2 |
-1 |
3(-1^{2}) – 6(-1) – 2 = -5 |
0 |
3(0^{2}) – 6(0) – 2 = -2 |
1 |
3(1^{2}) – 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.