### 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) = 2x4 + x2 is an even function.

f(x) = 2x4 + x2
f(-x) = 2(-x)4 + (-x)2
f(-x) = 2x4 + x2 = f(x)

Therefore, f(x) = 2x4 + x2 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) = x3 + 2x is an odd function.
f(x) = x3 + 2x
f(-x) = (-x)3 + 2(-x) = -x3 -2x
-f(-x) = x3 + 2x = f(x)

Therefore, f(x) = x3 + 2x is an odd function.

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

1. Show by drawing or otherwise that f(x) = 1/x and f(x) = x3 are odd functions.
2. Show that f(x) = (x - 2)(x + 2)/(x2 + 2) is an even function.
3. Check whether f(x) = x3 / (x2 + 7) is odd or even.
4. Show that f(x) = (x3 + x) /(x3 - x) is an even function.

### Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
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