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 below 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 below 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:

Interactive Exercises

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