**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.

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.

This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

~~"There's no such thing as a free lunch."~~

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Please use them and excel in the sphere of science education.

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Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7^{th} edition in print.