2.04 Odd and even functions

Functions can be classified in three ways. There are odd functions, there are even functions and there are functions which are neither odd nor even. The reason for these classifications of functions has to do with symmetry. Knowing that a function is odd or even assists us with understanding more about the shape of a function when graphed.  

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 show that:

$f\left(-x\right)$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 exhibits reflective symmetry across the $y$y-axis as shown in the graph below. 

 

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)

 

It is very important that we show all the steps of working shown above, including the factoristation, if we are asked to algebraically show that a function is odd. The factorisation is crucial in the proof of the property that $f(-x)=-f(x)$f(−x)=−f(x). 

As a graph, an odd function possesses rotational symmetry. Specifically, this means that an odd function can be rotated $180^\circ$180°  about the origin and fall 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 $x$x exhibits the property of an odd function, that is $f\left(-x\right)=-f\left(x\right)$f(−x)=−f(x). 

 

 

If a function isn't odd or even, then it is said to be neither, and many functions we encounter are in this last category. This is particularly true for functions that seem to have symmetry but not in the exact manner we are seeking. An example is the graph shown below. Its rotational symmetry isn't centred on the origin. Looking closely at the $y$y values of the curve, it is clear that $f(-x)\ne-f(x)$f(−x)≠−f(x). In fact, you might have realised the important condition that if a continuous function is going to be odd, it must pass through the origin.

 

 

Examples of odd and even functions

We can suspect a function is odd when the powers of $x$x 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 a function is even when the powers of $x$x 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, 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. 

 

 

Practice Questions

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QUESTION 3