Strand 5 — Functions

Concept of a Function

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓Understand what a function is and how it maps inputs to outputs.
✓Identify the domain, codomain, and range of a function.
✓Use f(x) notation to represent and evaluate functions.
✓Distinguish between a function and a relation that is not a function.

Key concepts

Function

A function is a special type of relation where each input (from the domain) is mapped to exactly one output (in the codomain). Think of it as a machine: you put something in, and exactly one thing comes out. If an input had more than one output, it would not be a function.

Domain

The domain of a function is the set of all possible input values (often represented by x-values) for which the function is defined. These are the values you are allowed to 'put into' the function machine.

Codomain

The codomain of a function is the set of all possible output values (often represented by y-values) that the function *could* produce. It's the 'target' set where the outputs are expected to land. The range is always a subset of the codomain.

Range

The range of a function is the set of all *actual* output values (y-values) that the function produces when all inputs from the domain are used. The range is always a subset of the codomain.

f(x) Notation

The notation f(x) is read as 'f of x' and represents the output of the function f when the input is x. It's a concise way to write the rule for a function. For example, if f(x) = 2x + 1, it means that for any input x, the function multiplies it by 2 and then adds 1.

Key facts to remember

1A function maps each input to exactly one output.
2The domain is the set of all input values.
3The codomain is the set of all *possible* output values.
4The range is the set of all *actual* output values.
5The range is always a subset of the codomain.
6f(x) notation means 'the value of the function f at x'.
7To evaluate f(a), substitute 'a' for 'x' in the function's rule.

Worked examples

Example 1

Given the function f(x) = 3x - 2, find f(4) and f(-1).

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

IIf(4) = 3(4) - 2

IIICalculate the value:

IVf(4) = 12 - 2 = 10

VTo find f(-1), substitute x = -1 into the function rule:

VIf(-1) = 3(-1) - 2

VIICalculate the value:

VIIIf(-1) = -3 - 2 = -5

Answer

f(4) = 10, f(-1) = -5

Example 2

A function g is defined by the mapping diagram below. Domain = {1, 2, 3, 4} Codomain = {2, 3, 4, 5, 6, 7} Mappings: 1 → 3, 2 → 4, 3 → 5, 4 → 6 Identify the domain, codomain, and range of the function g.

IThe domain is given as the set of all input values.

IIThe codomain is given as the set of all possible output values.

IIIThe range is the set of all actual output values produced by the function from the given domain.

Answer

Domain = {1, 2, 3, 4}, Codomain = {2, 3, 4, 5, 6, 7}, Range = {3, 4, 5, 6}

Notice that the range is a subset of the codomain. Not all elements of the codomain are necessarily in the range.

Example 3

Let the function h be defined by h(x) = x² + 1. If the domain is {-2, 0, 1}, find the range of h.

ISubstitute each value from the domain into the function to find the corresponding output.

IIFor x = -2: h(-2) = (-2)² + 1 = 4 + 1 = 5.

IIIFor x = 0: h(0) = (0)² + 1 = 0 + 1 = 1.

IVFor x = 1: h(1) = (1)² + 1 = 1 + 1 = 2.

VCollect all the actual output values to form the range.

Answer

Range = {1, 2, 5}

Remember that squaring a negative number results in a positive number, e.g., (-2)² = 4.

Common mistakes

✗Confusing the range with the codomain. The range is the *actual* outputs, while the codomain is the *possible* outputs.
✗Incorrectly evaluating f(x) for negative inputs, especially with squaring (e.g., thinking (-2)² is -4 instead of 4).
✗Assuming every relation is a function; remember the 'each input to exactly one output' rule.
✗Not listing all elements in the range when the domain is a set of discrete values.

Exam tips

★Always clearly state the domain, codomain, and range when asked, using correct set notation (curly brackets {}).
★When evaluating f(x), show your substitution step clearly to avoid calculation errors and earn partial marks.
★Practice identifying whether a given relation (e.g., from a graph or set of ordered pairs) is a function.
★Pay close attention to the specified domain; it dictates which inputs you should consider.

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