Section A — Chapters 1–22

Introduction to Functions

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to define what a function is.
✓By the end of this lesson students will be able to identify the domain, codomain, and range of a function given a set of inputs and outputs.
✓By the end of this lesson students will be able to understand and use function notation, such as f(x).
✓By the end of this lesson students will be able to evaluate a function for a given input value (object).
✓By the end of this lesson students will be able to find the input value (object) given an output value (image).

Key concepts

Function

A function is a special type of rule that takes an input value and assigns it to exactly one output value. Think of it like a machine: you put something in, and it gives you one specific thing out. For every input, there is only one output.

Object

An object is an input value to a function. It is what you 'put into' the function.

Image

An image is the output value that a function produces for a given object. It is what you 'get out' of the function.

Domain

The domain of a function is the set of all possible input values (objects) that the function can take.

Codomain

The codomain of a function is the set of all possible output values. It is the set where all the images are expected to be found. The range is always a subset of the codomain.

Range

The range of a function is the set of all actual output values (images) that the function produces when applied to the objects in its domain. It is a subset of the codomain.

Function Notation

Function notation is a way to write functions using symbols. The most common notation is f(x), which is read as 'f of x'. Here, 'f' is the name of the function, and 'x' represents the input value (object). So, f(x) represents the output value (image) of the function f when the input is x.

f(x) = expression

Key facts to remember

1A function assigns exactly one output to each input.
2The input values are called objects.
3The output values are called images.
4The domain is the set of all possible input values.
5The codomain is the set of all possible output values.
6The range is the set of all actual output values, and it is a subset of the codomain.
7Function notation f(x) means 'the value of function f at x'.
8To evaluate f(x) for a specific value, substitute that value for x in the function's rule.

Worked examples

Example 1

Given the function f(x) = 3x + 2, find the image of 4. In other words, find f(4).

IWrite down the function: f(x) = 3x + 2

IITo find f(4), substitute x = 4 into the function:

IIIf(4) = 3(4) + 2

IVPerform the multiplication: f(4) = 12 + 2

VPerform the addition: f(4) = 14

Answer

f(4) = 14

The object is 4, and its image is 14.

Example 2

A function g maps the set A = {1, 2, 3} to the set B = {2, 4, 6, 8} by the rule g(x) = 2x. Identify the domain, codomain, and range of this function.

IIdentify the domain: The domain is the set of all possible input values, which is given as set A.

IIDomain = {1, 2, 3}

IIIIdentify the codomain: The codomain is the set of all possible output values, which is given as set B.

IVCodomain = {2, 4, 6, 8}

VCalculate the images for each object in the domain to find the range:

VIFor x = 1, g(1) = 2(1) = 2

VIIFor x = 2, g(2) = 2(2) = 4

VIIIFor x = 3, g(3) = 2(3) = 6

9The set of actual output values is {2, 4, 6}. This is the range.

10Range = {2, 4, 6}

Answer

Domain = {1, 2, 3}, Codomain = {2, 4, 6, 8}, Range = {2, 4, 6}

Notice that the range is a subset of the codomain.

Example 3

For the function h(x) = x - 5, if the image is 10, what is the object? In other words, find x when h(x) = 10.

IWrite down the function and the given image: h(x) = x - 5 and h(x) = 10

IISet the function equal to the image: x - 5 = 10

IIITo find x, we need to isolate it. Add 5 to both sides of the equation:

IVx - 5 + 5 = 10 + 5

Vx = 15

Answer

The object is 15.

When the image is 10, the corresponding object (input) is 15.

Common mistakes

✗Confusing domain and range: Remember, domain is for inputs (objects), range is for outputs (images).
✗Assuming the codomain is always the same as the range: The range is only the *actual* outputs, while the codomain is the *possible* outputs. The range is a subset of the codomain.
✗Incorrectly substituting values: Make sure to replace *all* instances of 'x' with the input value when evaluating a function.
✗Not following the order of operations (BODMAS/PEMDAS) when evaluating functions.
✗Thinking a function can have multiple outputs for a single input. This is not a function.

Exam tips

★Always read the question carefully to identify what you are asked to find: an image, an object, the domain, or the range.
★Show all your working steps clearly, especially when substituting values into a function. This helps you get partial marks even if your final answer is incorrect.
★Use brackets when substituting negative numbers to avoid sign errors, e.g., f(-2) = 3(-2) + 2.
★If a mapping diagram is given, draw arrows from each domain element to its unique image to visualise the function and easily identify the range.

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