Review of Functions: Learn it 4

Symmetry of Functions

Function graphs often exhibit symmetry, a feature that can simplify understanding their behavior.

Symmetry about the [latex]y[/latex]-axis means that mirroring the graph over the [latex]y[/latex]-axis results in the same graph, indicating an even function where [latex]f(x)=f(−x)[/latex]. For instance, [latex]f(x)=x^4−2x^2−3[/latex] is even because both sides of the [latex]y[/latex]-axis mirror each other.

Symmetry about the origin implies that rotating the graph [latex]180[/latex] degrees around the origin leaves the graph unchanged. This is characteristic of odd functions, satisfying [latex]f(−x)=−f(x)[/latex]. Take [latex]f(x)=x^3−4x[/latex] as an example; it’s odd because rotating its graph doesn’t alter it.

Algebraically, you can check for y-axis symmetry by seeing if [latex]f(−x)[/latex] equals [latex]f(x)[/latex], and for origin symmetry by checking if [latex]f(−x)[/latex] equals [latex]−f(x)[/latex].

An image of two graphs. The first graph is labeled “(a), symmetry about the y-axis” and is of the curved function “f(x) = (x to the 4th) - 2(x squared) - 3”. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. This function decreases until it hits the point (-1, -4), which is minimum of the function. Then the graph increases to the point (0,3), which is a local maximum. Then the the graph decreases until it hits the point (1, -4), before it increases again. The second graph is labeled “(b), symmetry about the origin” and is of the curved function “f(x) = x cubed - 4x”. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. The graph of the function starts at the x intercept at (-2, 0) and increases until the approximate point of (-1.2, 3.1). The function then decreases, passing through the origin, until it hits the approximate point of (1.2, -3.1). The function then begins to increase again and has another x intercept at (2, 0).

Figure 13. (a) A graph that is symmetric about the [latex]y[/latex]-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 [latex]f[/latex] has symmetry? It becomes straightforward to identify symmetry in functions once we determine if they are even or odd. Even functions are symmetric about the y-axis, whereas odd functions exhibit symmetry about the origin.

even and odd functions

  • If [latex]f(-x)=f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], then [latex]f[/latex] is an even function. An even function is symmetric about the [latex]y[/latex]-axis.
  • If [latex]f(−x)=−f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], then [latex]f[/latex] is an odd function. An odd function is symmetric about the origin.

Determine whether each of the following functions is even, odd, or neither.

  1. [latex]f(x)=-5x^4+7x^2-2[/latex]
  2. [latex]f(x)=2x^5-4x+5[/latex]
  3. [latex]f(x)=\dfrac{3x}{x^2+1}[/latex]


Absolute Value Function

One symmetric function that arises frequently is the absolute value function, written as [latex]|x|[/latex]. The absolute value function is defined as

[latex]f(x)=\begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}[/latex]

 

Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if [latex]x<0[/latex], then [latex]|x|=−x>0[/latex], and if [latex]x>0[/latex], then [latex]|x|=x>0[/latex]. However, for [latex]x=0, \, |x|=0[/latex]. Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if [latex]x=0[/latex], the output [latex]|x|=0[/latex]. We can conclude that the range of the absolute value function is [latex]\{y|y\ge 0\}[/latex].

In Figure 14, we see that the absolute value function is symmetric about the [latex]y[/latex]-axis and is therefore an even function.

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -4 to 4. The graph is of the function “f(x) = absolute value of x”. The graph starts at the point (-3, 3) and decreases in a straight line until it hits the origin. Then the graph increases in a straight line until it hits the point (3, 3).
Figure 14. The graph of [latex]f(x)=|x|[/latex] is symmetric about the [latex]y[/latex]-axis.

Find the domain and range of the function [latex]f(x)=2|x-3|+4[/latex].