Properties of Functions and Basic Function Types
In this section, you will analyze graphs to determine whether they represent a function and be introduced to the graphs of the basic functions. Pay close attention to the basic functions because they will be referred to throughout most of the course.
Using the Horizontal Line Test
Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.
HOW TO
Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.
1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
2. If there is any such line, determine that the function is not one-to-one.
EXAMPLE 15
Applying the Horizontal Line Test
Consider the functions shown in Figure 9(a) and Figure 9(b). Are either of the functions one-to-one?
Solution
The function in Figure 9(a) is not one-to-one. The horizontal line shown in Figure 12 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points).
Figure 12
The function in Eigure 9(b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.
TRY IT #12
Is the graph shown in Figure 9 one-to-one?