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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Functions

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Contents 1 Introduction 1.1 Example 2 The inverse of a function 2.1 Example 3 Graphs 4 Composite Functions 4.1 Example 5 Quadratic Functions 6 The modulus function 7 Transforming graphs 8 Comments

Introduction

The phrase 'y is a function of x' means that the value of y depends upon the value of x, so y can be written in terms of x (e.g. ).

If , and y is a function of x (i.e. ), then the value of y when x is 4 is , which is found by replacing x's by 4's .

Example

If

Functions can be represented using a diagram. For example, the function :

(diagram for this example needed)

The domain is the set which the function is performed upon. The range is the set which contains the image of members of the domain. The range is a subset of the codomain. For example, the codomain may be the set of real numbers (all numbers you've come across). The range is the part of the codomain which have been mapped from the domain.

The inverse of a function

The inverse of a function is the function which reverses the effect of the original function. For example the inverse of is .

To find the inverse of a function, swap the x's and y's and make y the subject of the formula.

Example

Find the inverse of

Let , therefore

swap the x's and y's:

Make y the subject of the formula:

, so

Therefore

is the standard notation for the inverse of

Graphs

Functions can be graphed. When graphing functions, the domain will go on the x-axis, since this is the independent variable and the range will go on the y-axis.

A function is continuous if its graph has no breaks in it. An example of a discontinuous graph is :

(diagram for this example needed)

A function is periodic if its graph repeats itself at regular intervals, this interval being known as the period.

A function is even if it is unchanged when x is replaced by -x . The graph of such a function will be symmetrical in the y-axis. Even functions have even degrees (e.g. ).

A function is odd if the sign of the function is changed when x is replaced by -x . The graph of the function will have rotational symmetry about the origin (e.g. ).

Composite Functions

Composite functions are combinations of two or more functions.

(which is the same as or ), means do function first, then function .

Example

If and , find .

Quadratic Functions

The quadratic equation gives rise to the fact that real solutions will only exist if . The expression is therefore important, and is known as the discriminant.

(missing diagram)

A function is positive definite if it is always positive. For example . A quadratic function will be positive definite if and (i.e. the graph is u-shaped and does not cross the x-axis).

The modulus function

The modulus of a number is the magnitude of that number. For example, the modulus of -1 ( |-1| ) is 1. The modulus of x, |x|, is x for values of x which are positive and -x for values of x which are negative. So the graph of is for all positive values of x and for all negative values of x:

(graph of the modulus function needed)

Transforming graphs

If , the graph of (where c is a constant) will be the graph of shifted c units upwards (in the direction of the y-axis).

If , the graph of will be the graph of shifted c units to the left.

If , the graph of is a stretch of the graph of , scale factor a, from the x-axis.

Comments

This article is a good summary. It does not explain somethings for people who wish to know 'why' - it is ore suited for people who want ways to remember things, or a summary of what they need to know once they have been taught the topics properly with the 'why' bits.

The section on transforming graphs is very brief and has some key transformations needed in C2 (and possibly C1) missing.

The article is also lacking anything about odd and even functions, which are still needed at A Level and are possible things people have had difficultly picking up or which they may have forgotten from GCSE.