S4 Mathematics - Term 1

Topic 1: Composite Functions

Lesson 4: Functions Review and Transition to Composites

Duration: 40 minutes

Learning Outcomes

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

• Demonstrate fluency in evaluating functions
• Apply function notation to solve problems
• Understand applying one function after another

Review Quiz

Work individually in your exercise book:

Q1: Define a function in your own words.

Q2: Given f(x) = 4x - 1, find f(2) and f(5)

Q3: Given g(x) = x², find g(3) and g(-2)

Q4: If h(x) = 3x and h(a) = 15, find a

Quiz Answers

Q1: A function maps each input to exactly one output

Q2: f(2) = 7, f(5) = 19

Q3: g(3) = 9, g(-2) = 4

Q4: a = 5

Our Two Functions (from NCDC)

For this lesson:

f(x) = 6x (multiply by 6)

g(x) = x + 5 (add 5)

Practice Evaluating

f(x) = 6x:

• f(3) = 6(3) = 18
• f(5) = 6(5) = 30

g(x) = x + 5:

• g(3) = 3 + 5 = 8
• g(10) = 10 + 5 = 15

Real-World Problem: Akello's Pay

Akello earns UGX 6,000 per hour.
She also gets UGX 5,000 transport allowance daily.

f(x) = 6000x gives earnings for x hours
g(x) = x + 5000 adds the allowance

Calculating Akello's Pay

For 4 hours worked:

Step 1: Earnings = f(4) = 6000(4) = UGX 24,000

Step 2: Total with allowance = g(24000) = 24000 + 5000 = UGX 29,000

A New Question

What if we wanted to do both steps at once?

First find earnings (f), then add allowance (g)

Can we combine these functions?

The Key Curriculum Question

Given f(x) = 6x and g(x) = x + 5:

What is g(f(3))?

Breaking It Down

g(f(3)) means:

1. First, find f(3)
2. Then, use that result as input for g

Work from the inside out!

Step-by-Step Solution

Find g(f(3)):

Step 1: f(3) = 6(3) = 18

Step 2: g(18) = 18 + 5 = 23

Answer: g(f(3)) = 23

Visualizing with a Flow Diagram

Now Try: f(g(3))

Find f(g(3)):

Step 1: g(3) = 3 + 5 = 8

Step 2: f(8) = 6(8) = 48

Answer: f(g(3)) = 48

Compare the Results

• g(f(3)) = 23 (multiply first, then add)
• f(g(3)) = 48 (add first, then multiply)

The order matters!

Key Observation

g(f(3)) ≠ f(g(3))

23 ≠ 48

The order in which we apply functions changes the result!

Pair Practice

Given: f(x) = 2x and g(x) = x + 1

Find:

• g(f(4)) = ?
• f(g(4)) = ?

Compare your answers!

Solution: g(f(4))

Step 1: f(4) = 2(4) = 8

Step 2: g(8) = 8 + 1 = 9

g(f(4)) = 9

Solution: f(g(4))

Step 1: g(4) = 4 + 1 = 5

Step 2: f(5) = 2(5) = 10

f(g(4)) = 10

Compare Again

• g(f(4)) = 9
• f(g(4)) = 10

Again, different order gives different results!

The Rule: Inside First

For g(f(x)):

1. Evaluate the inside function first (f)
2. Use that result as input for the outside function (g)

Preview: Composite Functions

When we apply one function after another, we create a composite function.

g(f(x)) means: "apply f first, then g"

This is a key topic for UCE!

Summary: Lessons 1-4

Exit Questions

Given f(x) = 2x and g(x) = x - 3:

1. Find f(5)
2. Find g(10)
3. Find g(f(5))
4. Find f(g(5))

Exit Answers

1. f(5) = 2(5) = 10

2. g(10) = 10 - 3 = 7

3. g(f(5)) = g(10) = 10 - 3 = 7

4. f(g(5)) = f(2) = 2(2) = 4
   (since g(5) = 5 - 3 = 2)

Homework

Given f(x) = 3x and g(x) = x + 2:

1. Find f(4), g(6), f(-2), g(0)

2. Find g(f(2)) step by step

3. Find f(g(2)) step by step

4. Compare answers to Q2 and Q3

5. Draw a flow diagram for g(f(5))

Next Lesson

Lesson 5: Composite Functions

We will learn:

• Formal notation: g∘f
• How to write composite functions as formulas
• More practice with "inside first" approach

Credits

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

Source: National Curriculum Development Centre (NCDC), Uganda