Determine Whether a Functions is Even, Odd, or Neither
Some functions have symmetry, meaning their graphs remain unchanged when reflected. For example, reflecting the toolkit functions [latex]f(x) = x^2[/latex] or [latex]f(x) = |x|[/latex] horizontally across the y-axis will produce the same graph. We call these functions even functions because they are symmetric about the y-axis.
If the graph of [latex]f(x) = x^3[/latex] or [latex]f(x) = \dfrac{1}{x}[/latex] is reflected across both the x-axis and y-axis, the result is also the original graph.
These graphs are symmetric about the origin, and we call functions with this type of symmetry odd functions.
A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\left(x\right)={2}^{x}[/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\left(x\right)=0[/latex].
even and odd functions
A function is called an even function if for every input [latex]x[/latex]
[latex]f\left(x\right)=f\left(-x\right)[/latex]
The graph of an even function is symmetric about the [latex]y\text{-}[/latex] axis.
A function is called an odd function if for every input [latex]x[/latex]
[latex]f\left(x\right)=-f\left(-x\right)[/latex]
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 [latex]f\left(x\right)=f\left(-x\right)[/latex]. If it does, it is even.
Determine whether the function satisfies [latex]f\left(x\right)=-f\left(-x\right)[/latex]. If it does, it is odd.
If the function does not satisfy either rule, it is neither even nor odd.
Is the function [latex]f\left(x\right)={x}^{3}+2x[/latex] even, odd, or neither?
Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let’s begin with the rule for even functions.
Because [latex]-f\left(-x\right)=f\left(x\right)[/latex], this is an odd function.
Analysis of the Solution
Consider the graph of [latex]f[/latex]. Notice that the graph is symmetric about the origin. For every point [latex]\left(x,y\right)[/latex] on the graph, the corresponding point [latex]\left(-x,-y\right)[/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f[/latex], and the corresponding point [latex]\left(-1,-3\right)[/latex] is also on the graph.
