26
Chapter 1 | Functions and Graphs
Figure 1.13 (a) A graph that is symmetric about the y -axis. (b) A graph that is symmetric about the origin.
If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function f has symmetry? Looking at Figure 1.14 again, we see that since f is symmetric about the y -axis, if the point ( x , y ) is on the graph, the point (− x , y ) is on the graph. In other words, f (− x ) = f ( x ). If a function f has this property, we say f is an even function, which has symmetry about the y -axis. For example, f ( x ) = x 2 is even because f (− x ) = (− x ) 2 = x 2 = f ( x ). In contrast, looking at Figure 1.14 again, if a function f is symmetric about the origin, then whenever the point ( x , y ) is on the graph, the point (− x , − y ) is also on the graph. In other words, f (− x ) =− f ( x ). If f has this property, we say f is an odd function, which has symmetry about the origin. For example, f ( x ) = x 3 is odd because f (− x ) = (− x ) 3 =− x 3 =− f ( x ). Definition If f ( x ) = f (− x ) for all x in the domain of f , then f is an even function . An even function is symmetric about the y -axis. If f (− x ) =− f ( x ) for all x in the domain of f , then f is an odd function . An odd function is symmetric about the origin.
Example 1.10 Even and Odd Functions
Determine whether each of the following functions is even, odd, or neither. a. f ( x ) =−5 x 4 +7 x 2 −2
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online