3.4 Determining Even and Odd Functions
Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions f(x)=x2f(x)=x2 or f(x)=|x|f(x)=|x| will result in the original graph. We say that these types of graphs are symmetric about the y-axis. Functions whose graphs are symmetric about the y-axis are called even functions.
If the graphs off(x)=x3f(x)=x3orf(x)=1xf(x)=1x were reflected over both axes, the result would be the original graph, as shown in Figure 3-11.
We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function.
Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, f(x)=2xf(x)=2x is neither even nor odd. Also, the only function that is both even and odd is the constant function f(x)=0f(x)=0.
Even and Odd Functions
A function is called an even function if for every input xx
f(x)=f(−x)f(x)=f(−x)
The graph of an even function is symmetric about the y-y- axis.
A function is called an odd function if for every input xx
f(x)=−f(−x)f(x)=−f(−x)
The graph of an odd function is symmetric about the origin.
How To
Given the formula for a function, determine if the function is even, odd, or neither.
- Determine whether the function satisfies f(x)=f(−x)f(x)=f(−x). If it does, it is even.
- Determine whether the function satisfies f(x)=−f(−x)f(x)=−f(−x). If it does, it is odd.
- If the function does not satisfy either rule, it is neither even nor odd.
Example 1: Determining whether a Function is Even, Odd, or Neither
1. Is the function f(x)=x3+2xf(x)=x3+2x even, odd, or neither?
Analysis
Consider the graph of ff in Figure 3-12. Notice that the graph is symmetric about the origin. For every point (x,y)(x,y) on the graph, the corresponding point (−x,−y)(−x,−y) is also on the graph. For example, (1, 3) is on the graph of ff, and the corresponding point (−1,−3)(−1,−3) is also on the graph.
2. Is the function f(s)=s4+3s2+7f(s)=s4+3s2+7 even, odd, or neither?
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