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)=x3−x 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)=x3−x 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)=x4−8x2+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)=x5−2x3+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)=x6−2, $f\left(x\right)=x^2$f(x)=x2, and $f\left(x\right)=3x^8-5x^4$f(x)=3x8−5x4 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=√r2−x2 for some radius $r$r is even because $\sqrt{r^2-\left(-x\right)^2}=\sqrt{r^2-x^2}$√r2−(−x)2=√r2−x2.
Worked Examples
QUESTION 1
Consider the function $y=x^3-4x$y=x3−4x.
Complete the table of values.
$x$x $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Determine whether the function is odd, even or neither.
neither
Aeven
Bodd
C
QUESTION 2
Consider the graph below.
Find the value of $y$y when $x=4$x=4.
Find the value of $y$y when $x=-4$x=−4.
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
Areflecting across the line $y=x$y=x
Breflecting across the $x$x-axis
Creflecting across the $y$y axis
DDetermine whether the function is odd, even or neither.
odd
Aneither
Beven
C
QUESTION 3
Consider the function $f\left(x\right)=x^3-5x$f(x)=x3−5x.
Find $f\left(-x\right)$f(−x).
- Therefore, determine whether the function is odd, even or neither.
Even
ANeither
BOdd
C