### Odd and Even functions

#### Advanced maths - odd and even functions tutorial

**Even Functions**

If a function exists such that f(x) = f(-x), it is called an even function.

**E.g.**

f(x) = cos x

This animation shows f(x) = f(-x).

Therefore, f(x) = cos x is an even function.

**E.g.**

Show that f(x) = 2x^{4} + x^{2} is an even function.

f(x) = 2x^{4} + x^{2}

f(-x) = 2(-x)^{4} + (-x)^{2}

f(-x) = 2x^{4} + x^{2} = f(x)

Therefore, f(x) = 2x^{4} + x^{2} is an even function.

**Odd Functions**

If a function exists such that f(x) = -f(-x), it is called an odd function.

**E.g.**

f(x) =sin x

This animation shows f(x) = -f(-x).

Therefore, f(x) = sin x is an even function.

**E.g.**

Show that f(x) = x^{3} + 2x is an odd function.

f(x) = x^{3} + 2x

f(-x) = (-x)^{3} + 2(-x) = -x^{3} -2x

-f(-x) = x^{3} + 2x = f(x)

Therefore, f(x) = x^{3} + 2x is an odd function.

**Practice is the key to mastering maths; please visit this page, for more worksheets.**

Please work out the following questions to complement what you have just learnt.

- Show by drawing or otherwise that f(x) = 1/x and f(x) = x
^{3} are odd functions.
- Show that f(x) = (x - 2)(x + 2)/(x
^{2} + 2) is an even function.
- Check whether f(x) = x
^{3} / (x^{2} + 7) is odd or even.
- Show that f(x) = (x
^{3} + x) /(x^{3} - x) is an even function.