Functions
Bookmark this page

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. y = 3x ).
If f(x) = 3x, and y is a function of x (i.e. y = f(x) ), then the value of y when x is 4 is f(4), which is found by replacing x's by 4's .

Example:
If f(x) = 3x + 4
f(5) = 3(5) + 4 = 19
f(x + 1) = 3(x + 1) + 4 = 3x + 7

Functions can be represented using a diagram. For example, the function f(x) = 2x + 1 :

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 y = 2x is y = 1/2 x .
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 f(x) = 2x + 1
Let y = f(x), therefore y = 2x + 1
swap the x's and y's:
x = 2y + 1
Make y the subject of the formula:
2y = x - 1, so y = ½(x - 1)
Therefore f ¯¹(x) = ½(x - 1)
f(to the power of -1)(x) is the standard notation for the inverse of f(x)

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 y = 1/x :

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. y = x²).
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. y = x³).

Composite Functions
Composite functions are combinations of two or more functions.
fog(x) (which is the same as f [g(x)] or fg(x) ), means do function g first, then function f.

Example:
If f(x) = x² and g(x) = 2x + 1, find fog(x).
fog(x) = f(2x + 1) = (2x + 1)²

Quadratic Functions
The quadratic equation gives rise to the fact that real solutions will only exist if b² - 4ac is greater or equal to 0. The expression b² - 4ac is therefore important, and is known as the discriminant.

A function is positive definite if it is always positive. For example y = x² + 1 . A quadratic function will be positive definite if b² - 4ac < 0 and a > 0 (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 y = |x| is y = x for all positive values of x and y = -x for all negative values of x:

Transforming graphs
If y = f(x), the graph of y = f(x) + c (where c is a constant) will be the graph of y = f(x) shifted c units upwards (in the direction of the y-axis).
If y = f(x), the graph of y = f(x + c) will be the graph of y = f(x) shifted c units to the left.
If y = f(x), the graph of y = af(x) is a stretch of the graph of y = f(x), scale factor a, from the x-axis.

Other Notes in this Category

1. Algebraic Long Division
2. Functions
3. Indicies
4. Logarithms
5. Partial Fractions
6. Reduction to Linear Form
7. Sequences
8. Series
9. Set Theory
10. Simultaneous Equations
11. Surds
12. The Binomial Series

Didn't find this useful?

• Visit Coursework.Info for over 14,000 GCSE, A-Level and University Essays