S4 Mathematics - Term 1

Topic 1: Composite Functions

Lesson 2: Domain and Range with Function Notation

Duration: 40 minutes

Learning Outcomes

By the end of this lesson, you should be able to:

• Identify the domain and range of a function
• Determine the range from a function rule
• Understand why some functions have restricted domains

Quick Review: Lesson 1

f(x) = 3x means:

• Multiply the input by 3

Find f(4):

• f(4) = 3(4) = 12

Review: Domain and Range

From S2 Mappings:

Domain: The set of all input values

Range: The set of all output values

Mapping Diagram Review

Domain: {1, 2, 3}

1 → 3
2 → 6
3 → 9

Range: {3, 6, 9}

Today's Goal

Find domain and range using function notation

Instead of drawing diagrams!

Domain and Range Visualized

Example 1: Finding the Range

Given: f(x) = 2x, where x ∈ {1, 2, 3, 4, 5}

Domain is given: {1, 2, 3, 4, 5}

Task: Find the range

Solution: Evaluate Each Input

The Range

Range = {2, 4, 6, 8, 10}

The range is the set of all outputs.

Pair Practice

Given: g(x) = x + 3, where x ∈ {0, 1, 2, 3}

• Domain = {0, 1, 2, 3}
• Find the range

Work with your partner (2 minutes)

Solution: g(x) = x + 3

Range = {3, 4, 5, 6}

Practice Problem

Given: h(x) = 3x - 1, where x ∈ {1, 2, 3}

Find the range in your exercise book.

Answer: h(x) = 3x - 1

• h(1) = 3(1) - 1 = 2
• h(2) = 3(2) - 1 = 5
• h(3) = 3(3) - 1 = 8

Range = {2, 5, 8}

Restricted Domains

Sometimes, not all inputs are allowed.

The valid inputs form the natural domain.

Restriction 1: Division by Zero

f(x) = 12/x

Can we find f(3)?

• f(3) = 12/3 = 4 ✓

Can we find f(0)?

• f(0) = 12/0 = ??? ✗

Division by Zero is Undefined!

12 ÷ 0 = undefined

So for f(x) = 12/x:

x cannot be 0

Restriction 2: Real-World Context

A farmer has x chickens.
f(x) = 2x gives the number of chicken legs.

Can x be -3?

• No! You cannot have -3 chickens.

Can x be 2.5?

• No! You cannot have half a chicken.

Domain: Real-World

For the chicken function:

Domain: x must be a whole number ≥ 0

x ∈ {0, 1, 2, 3, 4, ...}

Example: Squaring Function

g(x) = x², where x ∈ {-2, -1, 0, 1, 2}

Find the range.

Solution: g(x) = x²

The Range of g(x) = x²

Outputs: 4, 1, 0, 1, 4

Range = {0, 1, 4}

Notice: No duplicates in sets!

Key Observation

Different inputs can give the same output:

• g(-2) = 4
• g(2) = 4

But the range only lists 4 once!

Summary: Finding the Range

Process:

1. Look at each value in the domain
2. Evaluate f(x) for each input
3. Collect all outputs in a set (no duplicates)

Summary: Key Vocabulary

Exit Questions

Given: f(x) = 4x - 2, where x ∈ {0, 1, 2, 3}

1. What is the domain?
2. Find f(0), f(1), f(2), f(3)
3. What is the range?

Answers to Exit Questions

1. Domain = {0, 1, 2, 3}

2. f(0) = -2, f(1) = 2, f(2) = 6, f(3) = 10

3. Range = {-2, 2, 6, 10}

Homework

1. f(x) = 2x + 1, x ∈ {0, 1, 2, 3, 4}

   • State the domain and find the range

2. g(x) = x², x ∈ {-3, -2, -1, 0, 1, 2, 3}

   • State the domain and find the range

3. h(x) = 24/x gives travel time. Why can't x = 0?

Next Lesson

Lesson 3: Evaluating Functions

We will practice with:

• More complex function rules
• Different types of functions
• Multiple operations

Credits

Created: December 2025
Based on: NCDC Lower Secondary Mathematics Syllabus (2019)

Source: National Curriculum Development Centre (NCDC), Uganda