Functions

Functions are mathematical building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies and for keeping aeroplanes in the air. Functions can take input from many variables, but always give the same output, unique to that function.

Functions also allow us to visualise relationships in terms of graphs, which are much easier to read and interpret than lists of numbers.

Some examples of functions include:

• Money as a function of time. You never have more than one amount of money at any time because you can always add everything to give one total amount. By understanding how your money changes over time, you can plan to spend your money sensibly. Businesses find it very useful to plot the graph of their money over time so that they can see when they are spending too much.

• Temperature as a function of various factors. Temperature is a very complicated function because it has so many inputs, including: the time of day, the season, the amount of clouds in the sky, the strength of the wind, where you are and many more. But the important thing is that there is only one temperature output when you measure it in a specific place.

• Location as a function of time. You can never be in two places at the same time. If you were to plot the graphs of where two people are as a function of time, the place where the lines cross means that the two people meet each other at that time.

This idea is used in logistics, an area of mathematics that tries to plan where people and items are for businesses.

DEFINITION: Function

A function is a mathematical relationship between two variables, where

every input variable has one output variable.

Dependent and independent variables

In functions, the x-variable is known as the input or independent variable, because its value can be chosen freely. The calculated y-variable is known as the output or dependent variable, because its value depends on the chosen input value.

Set notation

Examples:

{x : x ∈ R, x > 0}           The set of all x-values such that x is an element
of the set of real numbers and is greater than 0.
{y : y ∈ N, 3 < y ≤ 5}     The set of all y-values such that y is a natural number,
is greater than 3 and is less than or equal to 5
{z : z ∈ Z, z ≤ 100}          The set of all z-values such that z is an integer and
is less than or equal to 100.

Interval notation

It is important to note that this notation can only be used to represent an interval of real

numbers.

Examples:

(3; 11)               Round brackets indicate that the number is not
included. This interval includes all real numbers
greater than but not equal to 3 and less than but
not equal to 11.
(−∞;−2)            Round brackets are always used for positive and
negative infinity. This interval includes all real
numbers less than, but not equal to −2.
[1; 9)                 A square bracket indicates that the number is included.
This interval includes all real numbers
greater than or equal to 1 and less than but not
equal to 9.

Function notation

This is a very useful way to express a function. Another way of writing y = 2x + 1 is f(x) = 2x + 1. We say “f of x is equal to 2x + 1”. Any letter can be used, for example, g(x), h(x), p(x), etc.

1. Determine the output value:

“Find the value of the function for x = −3” can be written as: “find f(−3)”.

Replace x with −3:

f(−3) = 2(−3) + 1 = −5  ∴ f(−3) = −5

This means that when x = −3, the value of the function is −5.

2. Determine the input value:

“Find the value of x that will give a y-value of 27” can be written as: “find x if

f(x) = 27”.

We write the following equation and solve for x:

2x + 1 = 27

∴ x = 13

This means that when x = 13 the value of the function is 27.

Representations of functions

Functions can be expressed in many different ways for different purposes.

1. Words: “The relationship between two variables is such that one is always 5 less than the other.”

2. Mapping diagram:

Input:          Function:             Output:

−3                                                    -8           

  0                    x – 5                        -5

  5                                                      0

3. Table:

Input variable (x)       −3      0      5
Output variable (y)   −8     −5      0

4. Set of ordered number pairs: (−3;−8), (0;−5), (5; 0)

5. Algebraic formula: f(x) = x − 5

Example 1  If two functions are given as f(x) = 2x + 3, and g(x) = 3 − x2, then

(a) f(2) = 2 × 2 + 3 = 7

(b) f(−3) = 2 × (−3) + 3 = −6 + 3 = −3

(c) g(0) = 3 − (0)2 = 3

(d) g(4) = 3 − (4)2 = 3 − 16 = −13

The function f(x) = 2x + 3 in example 1 may be represented as a

flow diagram.

x → multiply by 2→ 2x → add 3→ 2x+3

From the diagram it is clear that the order of the operations cannot

be confused. First multiply by 2 and then add 3.

COMPOSITE FUNCTION

The function f(x) = 2x + 3, from example 1, is composed of two

simpler functions, i.e. multiply by 2 and add 3. If these two functions

are written as h : x 7! 2x and g : x 7! x + 3 then the composition of

these two functions is written gh (sometimes as goh or g(h(x))).

Example : If h : x → 2x² and g : x →  , find the composite

function gh.

Solution

 Applying first h and then g results in the composite function

gh =

This can best be seen by using flow diagrams.

x → square x² → multiply by 2→2 x² → add 5 → 2 x²+5 → Square root→

N.B.: The composition of two functions, fg, is NOT the same as the product of two functions.

Inverse Functions

If a function f maps m to n then the inverse function, written as f−1,

maps n to m.

Example : Find the inverse of the function h : x → .

Solution

First draw a flow diagram for the function.

x → multiply by 4→ 4x → subtract 3→ 4x−3→ divide by 2 →

Now draw a flow diagram, starting from the right, with each operation

replaced by its inverse.

 → divide by 4 → 2x+3 → add 3 → 2x → multiply by 2 → x

The inverse of h : x →  is thus h⁻¹ : x →  .

Reference:

1.     Handbook on Mathematics.

2.     Book of Everything in Mathematics- Grade 9-10

3.     Wikipedia

4.     Wolfram Maths World.