3 Odd and even functions
Example 7
Figure 23 shows graphs of several functions. They share a common property. Study the graphs and comment on any symmetry.
Figure 23
The graphs are all symmetrical about the $y$ axis.
Any function which is symmetrical about the $y$ axis, i.e. where the graph of the righthand part is the mirror image of that on the left, is said to be an even function . Even functions have the following property:
Key Point 4
Even Function
An even function is such that $f\left(x\right)=f\left(x\right)$ for all values of $x$ .
Key Point 4 is saying that the function value at a negative value of $x$ is the same as the function value at the corresponding positive value of $x$ .
Example 8
Show algebraically that $f\left(x\right)={x}^{4}+5$ is an even function.
Solution
We must show that $f\left(x\right)=f\left(x\right)$ .
$\phantom{\rule{2em}{0ex}}f\left(x\right)={\left(x\right)}^{4}+5={x}^{4}+5$
Hence $f\left(x\right)=f\left(x\right)$ and so the function is even. Check for yourself that $f\left(3\right)=f\left(3\right)$ .
Task!
Extend the graph in the solution box in order to produce a graph of an even function.
Task!
The following diagrams shows graphs of several functions. They share a common property. Study the graphs and comment on any symmetry.
There is rotational symmetry about the origin. That is, each curve, when rotated through $18{0}^{\∘}$ , transforms into itself.
Any function which possesses such symmetry $$ that is the graph of the right can be obtained by rotating the curve on the left through $18{0}^{\∘}$ about the origin $$ is said to be an odd function. Odd functions have the following property:
Key Point 5
Odd Function
An odd function is such that $f\left(x\right)=f\left(x\right)$ for all values of $x$ .
Key Point 5 is saying that the function value at a negative value of $x$ is minus the function value at the corresponding positive value of $x$ .
Example 9
Show that the function $f\left(x\right)={x}^{3}+4x$ is odd.
Solution
We must show that $f\left(x\right)=f\left(x\right)$ .
$$\begin{array}{rcll}f\left(x\right)& =& {\left(x\right)}^{3}+4\left(x\right)& \text{}\\ & =& {x}^{3}4x& \text{}\\ & =& \left({x}^{3}+4x\right)& \text{}\\ & =& f\left(x\right)\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}& \text{}\end{array}$$and so this function is odd. Check for yourself that $f\left(2\right)=f\left(2\right)$ .
Task!
Extend the graph in the solution box in order to produce a graph of an odd function.
Note that some functions are neither odd nor even ; for example $f\left(x\right)={x}^{3}+{x}^{2}$ is neither even nor odd.
The reader should confirm (with simple examples) that, ‘odd’ and ‘even’ functions have the following properties:
$$\begin{array}{rcll}\text{odd+odd}& =& \text{oddeven+even}=\text{evenodd+even}=\text{neither}& \text{}\\ \text{odd}\phantom{\rule{1em}{0ex}}\times \phantom{\rule{1em}{0ex}}\text{odd}& =& \text{even}\phantom{\rule{2em}{0ex}}\text{even}\phantom{\rule{1em}{0ex}}\times \phantom{\rule{1em}{0ex}}\text{even}\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\text{even}\phantom{\rule{2em}{0ex}}\text{odd}\phantom{\rule{1em}{0ex}}\times \phantom{\rule{1em}{0ex}}\text{even=odd}& \text{}\end{array}$$Exercises

Classify the following functions as odd, even or neither. If necessary sketch a graph to help you decide.
 $f\left(x\right)=6$ ,
 $f\left(x\right)={x}^{2}$ ,
 $f\left(x\right)=2x+1$ ,
 $f\left(x\right)=x$ ,
 $f\left(x\right)=2x$

The diagram below represents a heavy cable hanging under gravity from two points at the same height. Such a curve (shown as a dashed line), known as a catenary , is described by a mathematical function known as a hyperbolic cosine, $f\left(x\right)=coshx$ , discussed in HELM booklet 6.
A catenary
 Comment upon any symmetry.
 Is this function onetoone or manytoone?
 Is this a continuous or discontinuous function?
 State $\underset{x\to 0}{lim}coshx$ .

 even,
 even,
 neither,
 odd,
 odd

 function is even, symmetric about the $y$ axis,
 manytoone,
 continuous,
 1