S4 Mathematics - Term 1

Topic 1: Composite Functions

Lesson 1: Introduction to Function Notation

Duration: 40 minutes

Learning Outcomes

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

• Recall the definition of a function from S2 mappings
• Understand and use function notation f(x)
• Evaluate a function for a given input value

Review: What is a Mapping?

From S2, we learned about mapping diagrams:

• A mapping shows how elements in one set relate to another
• Uses arrows to show the relationship

Example: "Double each number"

1 → 2
2 → 4
3 → 6

Review: Domain and Range

Domain: The set of input values (starting set)

Range: The set of output values (ending set)

Domain: {1, 2, 3}
Range:  {2, 4, 6}

What Makes a Mapping a Function?

A function is a special type of mapping where:

Each input maps to exactly ONE output

Function: Each Input → One Output

NOT a Function: One Input → Two Outputs

Introducing Function Notation

Instead of drawing mapping diagrams, we use function notation:

f(x)

Read as: "f of x"

Understanding f(x)

f(x) = 2x

• f is the name of the function
• x is the input value
• 2x is the rule (multiply by 2)
• f(x) is the output

Function as a Machine

Important!

f(x) does NOT mean "f times x"

f(x) means:
"The value of function f when the input is x"

From Mapping to Notation

Mapping diagram:

1 → 2
2 → 4
3 → 6

Function notation:

f(x) = 2x

Both represent the same function!

Evaluating a Function

To find f(3) when f(x) = 2x:

1. Take the rule: 2x
2. Replace x with 3: 2(3)
3. Calculate: 6

Answer: f(3) = 6

Example: f(x) = 6x

From the NCDC curriculum:

What is f(3)?

f(3) = 6(3) = 18

Let's Practice Together

Given: f(x) = 6x

Find:

• f(1) = ?
• f(5) = ?
• f(10) = ?

Answers: f(x) = 6x

• f(1) = 6(1) = 6
• f(5) = 6(5) = 30
• f(10) = 6(10) = 60

Using Different Function Names

We can name functions with different letters:

• f(x) = 2x (doubling function)
• g(x) = x + 5 (adding 5 function)
• h(x) = 3x - 1 (multiply by 3, then subtract 1)

Example: g(x) = x + 5

Find g(3):

g(3) = 3 + 5 = 8

Find g(10):

g(10) = 10 + 5 = 15

Find g(0):

g(0) = 0 + 5 = 5

Functions with Two Operations

h(x) = 2x + 1

To find h(3):

1. Multiply by 2: 2(3) = 6
2. Add 1: 6 + 1 = 7

h(3) = 7

Remember BODMAS!

Pair Practice

Given: h(x) = 2x + 1

Work with your partner to find:

• h(0) = ?
• h(4) = ?
• h(10) = ?

(2 minutes)

Answers: h(x) = 2x + 1

• h(0) = 2(0) + 1 = 0 + 1 = 1
• h(4) = 2(4) + 1 = 8 + 1 = 9
• h(10) = 2(10) + 1 = 20 + 1 = 21

Common Mistake to Avoid

Wrong: h(4) = 2(4 + 1) = 2(5) = 10 ✗

Correct: h(4) = 2(4) + 1 = 8 + 1 = 9 ✓

Follow the order in the expression!

Summary: Key Vocabulary

Summary: The Connection

S2 Mapping Diagrams → S4 Function Notation

Both represent the same idea:

• A rule that takes an input
• Produces exactly one output

Exit Questions

Given: f(x) = 4x and g(x) = x - 2

Find:

1. f(5) = ?
2. g(7) = ?
3. f(0) = ?

Write your answers in your exercise book.

Answers to Exit Questions

1. f(5) = 4(5) = 20
2. g(7) = 7 - 2 = 5
3. f(0) = 4(0) = 0

Homework

Given these functions, find the values:

1. f(x) = 5x: Find f(2), f(4), f(7)
2. g(x) = x + 8: Find g(0), g(5), g(12)
3. h(x) = 3x - 1: Find h(1), h(3), h(10)

Expected time: 15-20 minutes

Next Lesson

Lesson 2: Domain and Range with Function Notation

We will explore:

• Finding the domain of a function
• Finding the range of a function
• Restrictions on domain

Credits

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

Source: National Curriculum Development Centre (NCDC), Uganda