⚙️ Functions

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

Understand function notation f(x) and f: x → y
Find outputs for given inputs (evaluate functions)
Identify domain and range
Work with composite functions fg(x) and gf(x)
Find inverse functions f⁻¹(x)

Quick start

A function is like a machine: you put a number in, it does something to it, and gives you a number out. We write f(x) to mean "the output when x goes in".

📖 Learn

1. Function Notation

A function takes an input, applies a rule, and produces an output. We use f(x) (read "f of x") to describe the rule:

f(x) = 2x + 3

This means "multiply the input by 2 then add 3". Other names for the function letter are g, h, etc.

Alternative notation: f: x ↦ 2x + 3 means exactly the same thing — x maps to 2x + 3.

Evaluating: To find f(4), substitute x = 4: f(4) = 2(4) + 3 = 11

Finding x: If f(x) = 15, solve 2x + 3 = 15 → x = 6

💡 f(x) is NOT f × x. It is a rule applied to x, like a machine label.

2. Domain and Range

The domain is the set of allowed input values. The range is the set of possible output values.

Example: f(x) = 2x + 3 with domain {1, 2, 3, 4}

Outputs (range): f(1)=5, f(2)=7, f(3)=9, f(4)=11 → range = {5, 7, 9, 11}

Domain restriction: Sometimes we say "x ≥ 0" or "x is an integer" to limit inputs.

💡 Think of domain as the ingredient list and range as the possible finished products from that ingredient list.

3. Mapping Diagrams

A mapping diagram shows how each input maps to exactly one output. For a function, each input has exactly ONE arrow going out of it.

One-to-one: Each input → unique output (e.g. f(x) = 2x + 3)

Many-to-one: Different inputs → same output (e.g. f(x) = x², where f(2) = f(−2) = 4)

One-to-many is NOT a function — each input must give exactly one output.

💡 Vertical line test: if a vertical line crosses a graph more than once, it is NOT a function.

4. Composite Functions

A composite function applies one function, then feeds the result into another.

fg(x) means "apply g first, then f to the result"

Example: f(x) = 2x + 1 and g(x) = x²

fg(x): First g(x) = x², then f(x²) = 2x² + 1

gf(x): First f(x) = 2x+1, then g(2x+1) = (2x+1)²

fg(3): g(3) = 9, then f(9) = 19

⚠️ fg(x) ≠ gf(x) in general! Order matters — apply right-to-left.

5. Inverse Functions

The inverse function f⁻¹(x) reverses what f does. If f(a) = b, then f⁻¹(b) = a.

ff⁻¹(x) = x and f⁻¹f(x) = x

How to find f⁻¹:

1. Write y = f(x) e.g. y = 3x − 2

2. Swap x and y: x = 3y − 2

3. Rearrange for y: y = (x + 2) ÷ 3

4. Write as f⁻¹(x) = (x + 2) / 3

⚠️ f⁻¹(x) does NOT mean 1/f(x). The −1 is NOT a power here.

💡 The graph of f⁻¹ is a reflection of f in the line y = x.

✏️ Worked Examples

Example 1 – Evaluating a Function

Given f(x) = 4x − 7, find: (a) f(3) (b) f(−2) (c) the value of x when f(x) = 13

Step 1 (a): Substitute x = 3: f(3) = 4(3) − 7 = 12 − 7 = 5

Step 2 (b): Substitute x = −2: f(−2) = 4(−2) − 7 = −8 − 7 = −15

Step 3 (c): Set f(x) = 13: 4x − 7 = 13 → 4x = 20 → x = 5

Example 2 – Domain and Range

g(x) = x² − 1, domain = {−2, −1, 0, 1, 2}. Find the range.

Step 1: g(−2) = 4 − 1 = 3

Step 2: g(−1) = 1 − 1 = 0

Step 3: g(0) = 0 − 1 = −1

Step 4: g(1) = 1 − 1 = 0 (same as g(−1))

Step 5: g(2) = 4 − 1 = 3 (same as g(−2))

Range: {−1, 0, 3} — list each value once even if reached twice.

Example 3 – Composite Functions

f(x) = 3x + 2 and g(x) = x − 5. Find: (a) fg(x) (b) gf(x) (c) fg(4)

Step 1 (a) fg(x): Apply g first: g(x) = x − 5

Step 2: Apply f to result: f(x − 5) = 3(x − 5) + 2 = 3x − 15 + 2 = 3x − 13

Step 3 (b) gf(x): Apply f first: f(x) = 3x + 2

Step 4: Apply g to result: g(3x + 2) = 3x + 2 − 5 = 3x − 3

Step 5 (c): fg(4) using fg(x) = 3x − 13: 3(4) − 13 = 12 − 13 = −1 ✓

💡 Or directly: g(4) = −1, then f(−1) = 3(−1) + 2 = −1 ✓

Example 4 – Inverse Functions

f(x) = 5x + 3. Find f⁻¹(x) and verify f⁻¹(18) works correctly.

Step 1: Write y = 5x + 3

Step 2: Swap x and y: x = 5y + 3

Step 3: Rearrange: x − 3 = 5y → y = (x − 3) / 5

Step 4: f⁻¹(x) = (x − 3) / 5

Verify: f(3) = 5(3) + 3 = 18, so f⁻¹(18) should give 3

Check: f⁻¹(18) = (18 − 3) / 5 = 15 / 5 = 3 ✓

🎨 Visualizer

⚙️ Function Machine

Enter coefficients for f(x) = ax + b and an input value.

a: b: x:

🗺️ Mapping Diagram Builder

Choose a function and domain to see the mapping.

🔗 Composite Function Calculator

f(x) = ax + b g(x) = cx + d. Calculates fg(x) and gf(x).

a: b: c: d: x:

🔄 Inverse Function Finder

Enter f(x) = ax + b. See f⁻¹(x) step-by-step.

a: b:

Exercise 1 – Evaluating Functions

f(x) = 3x + 2. Find the value of each expression.

Exercise 2 – Domain and Range

g(x) = 2x − 1. Domain given. Find the output (range value) for each input.

Exercise 3 – Composite Functions

f(x) = 2x + 1 g(x) = x − 3. Evaluate each composite.

Exercise 4 – Inverse Functions

Find f⁻¹(x) for each function, then evaluate as directed. Enter numerical answers.