Hong Kong
Stage 4 - Stage 5

# Odd and Even Functions

Lesson

Functions can be grouped into three varieties. There are odd functions, there are even functions and there are functions which are neither odd nor even.

The reason for the classification of oddness and evenness of a function has to do with symmetric properties. Knowing that a function is odd or even assists us with understanding the function's graph on the cartesian plane.

## Odd functions

Algebraically speaking, an odd function $f\left(x\right)$f(x) exhibits the property that, for all $x$x values in the domain, $f\left(-x\right)=-f\left(x\right)$f(x)=f(x)

So for example the function given by $f\left(x\right)=x^3-x$f(x)=x3x is odd because:

 $f\left(-x\right)$f(−x) $=$= $\left(-x\right)^3-\left(-x\right)$(−x)3−(−x) $=$= $-x^3+x$−x3+x $=$= $-\left(x^3-x\right)$−(x3−x) $=$= $-f\left(x\right)$−f(x)

As a graph, an odd function possesses rotational symmetry. Specifically, an odd function, when rotated $180^\circ$180°  about the origin, falls back onto itself. The graph of $f\left(x\right)=x^3-x$f(x)=x3x is shown below. Note, from the example depicted in the diagram that any specific value of x in the domain exhibits the oddness property that $f\left(-x\right)=-f\left(x\right)$f(x)=f(x)

## Even functions

An even function $f\left(x\right)$f(x) exhibits the property that, for all $x$x values in the domain,  $f\left(-x\right)=f\left(x\right)$f(x)=f(x)

For example the function $f\left(x\right)=x^4-8x^2+16$f(x)=x48x2+16 is even because we can write:

 $f\left(-x\right)$f(−x) $=$= $\left(-x\right)^4-8\left(-x\right)^2+16$(−x)4−8(−x)2+16 $=$= $x^4-8x^2+16$x4−8x2+16 $=$= $f\left(x\right)$f(x)

An even function also possesses an interesting symmetry. An even function exhibits reflective symmetry across the y- axis as shown in the graph below.

If a function isn't odd or even, then it is said to be neither, and many functions we encounter are in this last catagory.

#### Examples of Odd and Even functions

We can suspect oddness when the powers of $x$x in the function are all odd. For example, $f\left(x\right)=x^3$f(x)=x3, and $f\left(x\right)=x^5-2x^3+7x$f(x)=x52x3+7x are all odd. We suspect even ness when the powers of $x$x in the function are all even. For example, $f\left(x\right)=x^6-2$f(x)=x62$f\left(x\right)=x^2$f(x)=x2, and $f\left(x\right)=3x^8-5x^4$f(x)=3x85x4 are all even.

But there are other functions to consider beside polynomials. For example, $y=\sin x$y=sinx is odd because $\sin\left(-x\right)=-\sin x$sin(x)=sinx and $y=\cos x$y=cosx is even because $\cos\left(-x\right)=\cos x$cos(x)=cosx.

Also the hyperbola $y=\frac{k}{x}$y=kx (for some constant $k$k) is odd because, for all $x$x in the domain,  $\frac{k}{\left(-x\right)}=-\frac{k}{x}$k(x)=kx. The semicircle $y=\sqrt{r^2-x^2}$y=r2x2 for some radius $r$r is even because $\sqrt{r^2-\left(-x\right)^2}=\sqrt{r^2-x^2}$r2(x)2=r2x2

#### Worked Examples

##### QUESTION 1

Consider the function $y=x^3-4x$y=x34x.

1. Complete the table of values.

 $x$x $y$y $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Determine whether the function is odd, even or neither.

neither

A

even

B

odd

C

##### QUESTION 2

Consider the graph below.

1. Find the value of $y$y when $x=4$x=4.

2. Find the value of $y$y when $x=-4$x=4.

3. How can the part of the graph for $x<0$x<0 be obtained by the part of the graph for $x>0$x>0?

rotating $180^\circ$180° about the origin

A

reflecting across the line $y=x$y=x

B

reflecting across the $x$x-axis

C

reflecting across the $y$y axis

D
4. Determine whether the function is odd, even or neither.

odd

A

neither

B

even

C

##### QUESTION 3

Consider the function $f\left(x\right)=x^3-5x$f(x)=x35x.

1. Find $f\left(-x\right)$f(x).

2. Therefore, determine whether the function is odd, even or neither.

Even

A

Neither

B

Odd

C