Functions & Graphing

Example Consider what happens to the number 7 in the two operations below: The number 7 is first multiplied by 4 and then 1 is added to the result to get 29. In this operation, 7 is called the input number and 29 is called the output number.

Function: y = 4x + 1 or f(x) = 4x + 1 The rule for the operation is ‘multiplied by 4 and add 1’. This is generally referred to as the function. If we call the input number x and the output number y we can write the ‘rule’ in terms of x & y. Function: y = 4x + 1 or f(x) = 4x + 1 This may also be written as f: x  4x + 1

Example Consider the function y = 5x – 4. Input all the numbers from the set {1, 2, 3, 4, 5, 6} The output numbers are {1, 6, 11, 16, 21, 26} The set of input numbers is called the domain. The set of output numbers is called the range. The input and output numbers can be represented in a mapping diagram. Each input number is mapped onto its output number.

Function A function is a rule that produces one output value only for each input value. Not a function since the element b is paired with 2 different elements in the range Is a function since each element in the domain is mapped onto one and only one element in the range.

Couples From the mapping diagram on previous page it can be seen that a function may be written as a set of couples or ordered pairs. i.e. (input, output). When a function is written as a set of couples, no two distinct couples will have the same input. {(a, d), (b, e), (b, f), (c, f)} is not a function since the input b has two different outputs. {(1, 1), (2, 4), (–2, 4)} is a functions since no two couples have the same input.

Codomain The set of inputs is called the domain. The set of outputs is called the range. The set of possible outcomes is called the codomain. Example: A = {1,2,3}, B = {1,3,5,7,9,11} and f(x) = 2x – 1. The couples are (1,1), (2,3) and (3,5). Looking at the mapping diagram for A and B, not all of the outputs are ‘used’. Therefore the range of this is {1,3,5}. However the full set B is known as the codomain.