The Basics of Functions Revision Notes for Leaving Cert Mathematics

The Basics of Functions

What is a function?

A function is like a machine that takes an input number, follows a specific rule, and produces an output number. Think of it as a mathematical recipe that always gives the same result when you put in the same ingredient.

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Key Definition: In mathematics, we use the word function for any rule that produces one output value only for each input value.

Function machines

Function machines (also called flow charts) show us visually how functions work. They consist of:

Input: The number that goes into the machine
Rule: The operations performed on the input
Output: The result that comes out

For example, consider this function machine:

7 → [multiply by 4] → [add 1] → 29

Here, 7 is the input, the rule is "multiply by 4 and add 1", and 29 is the output.

If we use x for the input and y for the output, we can write this rule as:

$y = 4x + 1$ , or
$x \to 4x + 1$

Real-life examples of functions

Functions are everywhere in daily life:

A calculator: input 6, press $x^2$ , output is 36
A barcode scanner: scan a book's barcode (input), the price appears (output)
Farming: apply fertiliser (input), crop yield increases (output)

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These examples help us understand that functions are not just abstract mathematical concepts - they represent relationships we encounter every day where one thing consistently leads to another specific result.

Domain, range and mapping diagrams

Key definitions

Domain: The set of all possible input values
Range: The set of all actual output values
Mapping diagram: A visual way to show how inputs connect to outputs

Understanding mapping diagrams

Consider the function machine: [multiply by 5] → [subtract 4]

If we input the numbers {1, 2, 3, 4, 5, 6}, we get outputs {1, 6, 11, 16, 21, 26}.

This can be shown in a mapping diagram:

Domain          Range
(Input)         (Output)
   1   --------→   1
   2   --------→   6  
   3   --------→   11
   4   --------→   16
   5   --------→   21
   6   --------→   26

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The Function Rule: In the mapping diagram, notice that each input has exactly one output. This is the key rule for functions.

Identifying functions

When looking at mapping diagrams, a relationship is a function if and only if:

Each element in the domain maps to one and only one element in the range

Examples of functions vs non-functions

Function example: {(1,4), (2,5), (3,6), (4,7)} ✓

Each input (1,2,3,4) has exactly one output

Non-function example: {(2,7), (3,8), (3,9), (4,12)} ✗

Input 3 has two different outputs (8 and 9), so this is not a function

Couples and ordered pairs

Functions can be written as sets of couples or ordered pairs in the form (input, output).

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Rules for couples

When a function is written as couples:

No two distinct couples will have the same input
The first number in each pair is the input
The second number in each pair is the output

Example: {(1,4), (2,5), (3,6), (4,7)} represents a function because no input appears twice.

Function notation

There are three common ways to write the same function rule:

The three notations

For the rule "double the number and add 4":

f(x) = 2x + 4 (function notation)
f: x → 2x + 4 (mapping notation)
y = 2x + 4 (equation form)

All three tell us that if the input is 3, the output is $2(3) + 4 = 10$ . We can write this as f(3) = 10.

Using function notation

If $f(x) = 2x + 4$ and we want to find $f(3)$ :

Substitute $x = 3$ into the rule
$f(3) = 2(3) + 4 = 6 + 4 = 10$

The codomain

Understanding the difference between these three important concepts:

Domain: The set of inputs = {1, 2, 3}
Range: The set of actual outputs = {1, 3, 5}
Codomain: The set of possible outputs = {1, 3, 5, 7, 9, 11}

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Key Point: The range is always a subset of the codomain. The codomain includes all possible outputs, while the range only includes the outputs that actually occur.

Worked example

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Worked Example: Finding Domain, Range and Couples

Question: A function is defined as $f: x \to 3x - 2$ . The domain of $f$ is {0, 1, 2, 3, 4}. Represent $f$ on a mapping diagram and write out the couples generated. What is the range of $f$ ?

Solution:

Step 1: Calculate outputs for each input

When $x = 0$ : $f(0) = 3(0) - 2 = -2$
When $x = 1$ : $f(1) = 3(1) - 2 = 1$
When $x = 2$ : $f(2) = 3(2) - 2 = 4$
When $x = 3$ : $f(3) = 3(3) - 2 = 7$
When $x = 4$ : $f(4) = 3(4) - 2 = 10$

Step 2: Create mapping diagram

Domain          Range
   0   --------→  -2
   1   --------→   1
   2   --------→   4  
   3   --------→   7
   4   --------→  10

Step 3: Write couples The couples are: {(0, -2), (1, 1), (2, 4), (3, 7), (4, 10)}

Step 4: Identify the range
The range is {-2, 1, 4, 7, 10}

Exam tips

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Important Exam Strategies

Always check that each input has exactly one output when identifying functions
Remember that domain = inputs, range = actual outputs, codomain = possible outputs
Practice all three function notations: $f(x) = ...$ , $f: x \to ...$ , and $y = ...$
When drawing mapping diagrams, use arrows to show the input-output relationships clearly
In couples, the first number is always the input (x-value)

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Key Points to Remember:

A function produces exactly one output for each input
The domain is the set of all inputs, the range is the set of all actual outputs
Mapping diagrams visually show how inputs connect to outputs with arrows
Functions can be written as couples like (input, output) or using function notation like $f(x) = 2x + 4$
The codomain contains all possible outputs, while the range contains only the actual outputs that occur

The Basics of Functions (Leaving Cert Mathematics): Revision Notes