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

Answer

Any function which is symmetrical about the $y$ axis, i.e. where the graph of the right-hand 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 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 $f\left(-x\right)=f\left(x\right)$ .

$f\left(-x\right)={\left(-x\right)}^4+5=x^4+5$

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.

Answer

Task!

The following diagrams shows graphs of several functions. They share a common property. Study the graphs and comment on any symmetry.

Answer

Any function which possesses such symmetry $-$ that is the graph of the right can be obtained by rotating the curve on the left through $180^{\∘}$ about the origin $-$ is said to be an odd function. Odd functions have the following property:

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 $f\left(-x\right)=-f\left(x\right)$ .

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.

Answer

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:

Exercises

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

2. 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

   1. Comment upon any symmetry.
   2. Is this function one-to-one or many-to-one?
   3. Is this a continuous or discontinuous function?
   4. State $\underset{x\to 0}{lim}coshx$ .

Answer