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 yy-axis means that mirroring the graph over the yy-axis results in the same graph, indicating an even function where f(x)=f(x)f(x)=f(x). For instance, f(x)=x42x23f(x)=x42x23 is even because both sides of the yy-axis mirror each other.

Symmetry about the origin implies that rotating the graph 180180 degrees around the origin leaves the graph unchanged. This is characteristic of odd functions, satisfying f(x)=f(x)f(x)=f(x). Take f(x)=x34xf(x)=x34x 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 f(x)f(x) equals f(x)f(x), and for origin symmetry by checking if f(x)f(x) equals f(x)f(x).

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 yy-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 ff 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 f(x)=f(x)f(x)=f(x) for all xx in the domain of ff, then ff is an even function. An even function is symmetric about the yy-axis.
  • If f(x)=f(x)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.

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

  1. f(x)=5x4+7x22
  2. f(x)=2x54x+5
  3. f(x)=3xx2+1


Absolute Value Function

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

f(x)={x,x0x,x<0

 

Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if x<0, then |x|=x>0, and if x>0, then |x|=x>0. However, for x=0,|x|=0. Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if x=0, the output |x|=0. We can conclude that the range of the absolute value function is {y|y0}.

In Figure 14, we see that the absolute value function is symmetric about the y-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 f(x)=|x| is symmetric about the y-axis.

Find the domain and range of the function f(x)=2|x3|+4.