Lesson 2–1

Example 1 Domain and Range	
State the domain and range of the relation 
shown in the graph. Is the relation a function?

The relation is {(–2, 2), (0, –4), (0, 3), (1, 2), (2, 3)}.
The domain is {–2, 0, 1, 2}.
The range is {–4, 2, 3}.

The relation is not a function because 0 is paired with both –4 and 3 in the range.v

Example 2  Vertical Line Test
Graph the values for the relation given in the table and determine whether it represents a function. Is the relation discrete or continuous?	x	y	
	0.50	2.50	
	1.00	3.00	
	0.50	1.00	
	1.75	
x

y

0

1

2

1

2

3

4

3.50	

	Use the vertical line test. Notice that there are two y-values for the x-value of 0.50. A vertical line intersects the graph in two points. Therefore, the graph does not represent a function. Because the graph consists of distinct points, the relation is discrete.	

Example 3  Graph a Relation
Graph each equation and find the domain and range. Then determine whether the equation is a function and state whether it is discrete or continuous.
a.	y = –x + 6
Make a table of values to find ordered pairs that satisfy the equation. Choose values for x and find the corresponding values for y. Then graph the ordered pairs.

	x	y		x	y	
		–2				–2	8	
		–1				–1	7	
		0				0	6	
		1				1	5	

Since x can be any real number, there are an infinite number of ordered pairs that can be graphed. All of them lie on the line shown. Notice that every real number is the x-coordinate of some point on the line. Also, every real number is the y-coordinate of some point on the line. So the domain and range are both all real numbers, and the relation is continuous.

This graph passes the vertical line test. For each x-value, there is exactly one y-value, so the equation y = -x + 6 represents a function.

b.	y = –x2 + 1
Make a table. Choose values for x and find the corresponding values for y. Then sketch the graph, connecting the points with a smooth curve.

	x	y		x	y	
		–2				–2		–3	
		–1				–1		0	
		0				0		1	
		1				1		0	
		2				2		3	

Since x can be any real number, there are an infinite number of ordered pairs that can be graphed. But, only real numbers less than or equal to 1 are y–coordinates of points on the graph. So the domain is all real numbers and the range is {y | y ≤ 1}.

You can see by examining the graph that no vertical line will intersect the graph in more than one point. For each x-value, there is exactly one y-value, so the equation y = -x2 + 1 represents a function.

Example 4  Evaluate a Function
Given f(x) = x3 – 5, find each value.
a.	f(–2)	b.	f(2a)
f(x) = x3 – 5	Original function			 f(x) = x3 – 5	Original function	
 f(–2) = (–2)3 – 5	Substitute.		f(2a) = (2a)3 – 5	Substitute.	
= –8 – 5 or –13	Simplify.		 = 8a3 – 5	(ab)x = axbx