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Functions
Complete Guide to Functions: IGCSE 0580 Mathematics
Domain: x ∈ ℝ, x ≠ −2.
2. Swap: x =
3. Solve: 5x = 3(y + 4)
5x = 3y + 12
5x − 12 = 3y
y =
4. g⁻¹(x) =
=
Welcome to this comprehensive guide on functions for IGCSE Mathematics (0580). This post covers everything you need to know about functions, including function notation, inverse functions, composite functions, and exponential functions - all essential topics for your exams.
1. Functions, Domain, Range, and Notation
Understanding the Basics
Function: A relation where every input (x-value) has exactly one output (y-value). Think of it as a machine: you put one number in, and get one specific number out.
Domain: The set of all possible input values (x-values) for the function.
Range: The set of all possible output values (y-values) that result from using the domain.
Notation: A function is usually written as f(x), read as "f of x". The 'f' is the function's name, and 'x' is the input variable. Other letters like g, h, p can also be used.
Example 1: Finding Outputs
Given f(x) = 3x − 5, find f(2) and f(−1).
Solution:
- f(2) = 3(2) − 5 = 6 − 5 = 1
- f(−1) = 3(−1) − 5 = −3 − 5 = −8
Example 2: Domain Considerations
Find the domain for:
a) f(x) = 3x − 5
b) p(x) =
3x + 2
Solution:
a) f(x) = 3x − 5 is a simple linear function. You can input any real number.
Domain: All real numbers, x ∈ ℝ.
b) p(x) =
3x + 2
is a fraction. The denominator cannot be zero, so x + 2 ≠ 0.Domain: x ∈ ℝ, x ≠ −2.
2. Inverse Functions f⁻¹(x)
Key Concepts
Inverse Function f⁻¹(x): A function that "reverses" the action of the original function f(x). If f(a) = b, then f⁻¹(b) = a.
How to find the inverse:
- Replace f(x) with y.
- Swap every x and y in the equation.
- Solve this new equation for y.
- Replace y with f⁻¹(x).
Example 1: Linear Function
Find the inverse of f(x) = 3x − 5.
Solution:
- y = 3x − 5
- Swap x and y: x = 3y − 5
- Solve for y:
x + 5 = 3y
y =x + 53 - f⁻¹(x) = x + 53
Example 2: Function with a Fraction
Find the inverse of f(x) =
3x + 2
, x ≠ −2.
Solution:
- y = 3x + 2
- Swap x and y: x = 3y + 2
- Solve for y:
x(y + 2) = 3
xy + 2x = 3
xy = 3 − 2x
y =3 − 2xx - f⁻¹(x) = 3 − 2xx, x ≠ 0
3. Composite Functions gf(x)
Understanding Composition
Composite Function: Combining two functions where the output of one function becomes the input of the other.
Notation: gf(x) means g(f(x)). You work from the inside out.
Important: Order matters! gf(x) is not usually the same as fg(x).
Example 1: Basic Composition
Given f(x) = 3x − 5 and g(x) =
3(x + 4)5
, find gf(2).
Solution:
- First, find the inner function f(2): f(2) = 3(2) − 5 = 1
- Now, use this result as the input for g(x): gf(2) = g(f(2)) = g(1)
- g(1) = 3(1 + 4)5=3 × 55= 3
- So, gf(2) = 3
Example 2: Finding a General Expression
Given f(x) =
3x + 2
and g(x) = (3x + 5)², find fg(x). Give your answer as a simplified fraction.
Solution:
- fg(x) = f(g(x)). So, the entire function g(x) becomes the input for f(x).
- In f(x), the input is x. We replace this input with g(x), which is (3x + 5)².
- fg(x) = f(g(x)) = 3(3x + 5)² + 2
- This is already a fraction in its simplest form.
4. Exponential Functions and Their Inverses
Working with Exponents
Exponential Functions: Functions where the variable is in the exponent, e.g., h(x) = 2ˣ, h(x) = 3ˣ
Key Properties:
- a⁰ = 1 for any a ≠ 0
- a¹ = a
- a⁻ⁿ = 1aⁿ
Example 1: Evaluating Exponential Functions
Given h(x) = 2ˣ, find:
a) h(3)
b) h(0)
c) h(−2)
Solution:
- h(3) = 2³ = 8
- h(0) = 2⁰ = 1
- h(−2) = 2⁻² = 12²=14
Example 2: Finding Inverse of Exponential Functions
Find the inverse of h(x) = 3ˣ.
Solution:
- y = 3ˣ
- Swap x and y: x = 3ʸ
- Solve for y: Since we cannot use logs at IGCSE level, we express this as:
If x = 3ʸ, then we can say y is the power we raise 3 to get x - h⁻¹(x) = the power such that 3 raised to that power equals x
5. Important Function Properties
Key Relationships
Function and Inverse Relationship:
- f(f⁻¹(x)) = x
- f⁻¹(f(x)) = x
Self-Inverse Functions: Functions where f⁻¹(x) = f(x)
Identity Property: h(h⁻¹(x)) = x for any function h
Example: Verifying Properties
Given f(x) = 2x + 3, verify that f(f⁻¹(x)) = x
Solution:
- First find f⁻¹(x):
1. y = 2x + 3
2. Swap: x = 2y + 3
3. Solve: x − 3 = 2y, so y =x − 32
4. f⁻¹(x) =x − 32 - Now find f(f⁻¹(x)):
f(f⁻¹(x)) = f(x − 32) = 2(x − 32) + 3 = (x − 3) + 3 = x ✓
6. Advanced Algebraic Manipulation with Functions
Complex Operations
Combining Functions: When adding/subtracting functions with different denominators, find a common denominator
Squaring Functions: (f(x))² means square the entire function, not just the x
Example 1: Writing as Single Fraction
Given f(x) =
3x+2
and g(x) = x, express f(x) + g(x) + 1 as a single fraction.
Solution:
- f(x) + g(x) + 1 = 3x+2+ x + 1
- Common denominator: x+2
- = 3x+2+x(x+2)x+2+1(x+2)x+2
- = 3 + x(x+2) + (x+2)x+2
- = 3 + x² + 2x + x + 2x+2
- = x² + 3x + 5x+2
Example 2: Complex Composite Pattern
Given g(x) = x² + 1, find gg(x) in the form ax⁴ + bx² + c
Solution:
- gg(x) = g(g(x)) = g(x² + 1)
- = (x² + 1)² + 1
- = (x⁴ + 2x² + 1) + 1
- = x⁴ + 2x² + 2
- So a = 1, b = 2, c = 2
Practice Exercises
Exercise 1: Basic Function Notation
1. For h(x) = 2x² + 3, find h(0) and h(3).
Show Answer
h(0) = 2(0)² + 3 = 3
h(3) = 2(3)² + 3 = 2(9) + 3 = 21
2. For k(x) =
5xx−4
, state a value that is not in the domain.
Show Answer
x = 4 is not in the domain because it makes the denominator zero.
Exercise 2: Inverse Functions
1. Find the inverse of g(x) =
3(x + 4)5
.
Show Answer
1. y =
3(x + 4)5
2. Swap: x =
3(y + 4)5
3. Solve: 5x = 3(y + 4)
5x = 3y + 12
5x − 12 = 3y
y =
5x − 123
4. g⁻¹(x) =
5x − 123
Exercise 3: Composite Functions
1. Using f(x) = 3x − 5 and g(x) =
3(x + 4)5
, find fg(x). Simplify your answer.
Show Answer
fg(x) = f(g(x)) = 3[g(x)] − 5 = 3[
3(x + 4)5
] − 5=
9(x + 4)5
− 5 = 9x + 365
− 255
= 9x + 36 − 255
= 9x + 115
Key Takeaways
- Always check domain restrictions, especially with fractions
- Remember the step-by-step process for finding inverse functions
- Work from the inside out when dealing with composite functions
- Practice identifying function properties and relationships
- Master exponential functions and their inverses for exam success
📐 Function rules – IGCSE Maths
⭐ Everything you need: inverse, composite, even/odd, transformations & more — made simple
🔄
Inverse function rules
f⁻¹(y) = x ⇔ f(x) = y💡 Idea: The inverse “undoes” the original function. If \(f(x)=y\), then \(f^{-1}(y)=x\).
✨ Key rules:
\(f^{-1}(f(x)) = x\) | \(f(f^{-1}(y)) = y\)
Domain of \(f^{-1}\) = Range of \(f\)
\(f^{-1}(f(x)) = x\) | \(f(f^{-1}(y)) = y\)
Domain of \(f^{-1}\) = Range of \(f\)
🎯 Example:
\(g(x)=2x+3\)
\(g^{-1}(7)=?\) → \(2x+3=7\) → \(x=2\)
✅ \(g^{-1}(7)=2\)
\(g(x)=2x+3\)
\(g^{-1}(7)=?\) → \(2x+3=7\) → \(x=2\)
✅ \(g^{-1}(7)=2\)
📌 \(g^{-1}(x) = \frac{x-3}{2}\)
🧩
Composite function rules
(f ∘ g)(x) = f(g(x))🔁 Order matters! First apply the inner function, then the outer one.
🌱 Example: \(f(x)=x^2,\; g(x)=x+1\)
\((f \circ g)(x)=(x+1)^2\)
\((g \circ f)(x)=x^2+1\)
⚠️ Usually \(f \circ g \neq g \circ f\)
\((f \circ g)(x)=(x+1)^2\)
\((g \circ f)(x)=x^2+1\)
⚠️ Usually \(f \circ g \neq g \circ f\)
📘 IGCSE tip: Write the inner substitution clearly, then simplify.
🎯 \( (f \circ g)(x) = f(g(x)) \)
🔄 Inverse Functions – Quick Notes
One simple rule • No finding inverses • Works for any function
⭐
The only rule you need
\( f^{-1}(y) = x \) means exactly the same as \( f(x) = y \)
That’s it. No long steps. No solving for \( f^{-1}(x) \). Just turn inverse questions into normal function questions.
📖
What it looks like
| If you see ... | It means ... | What to do |
|---|---|---|
| \( f^{-1}(5) = 2 \) | \( f(2) = 5 \) | Put 2 into \( f \) |
| \( f^{-1}(x) = 3 \) | \( f(3) = x \) | Find \( f(3) \) |
| \( f^{-1}(7) = x \) | \( f(x) = 7 \) | Solve \( f(x) = 7 \) |
🧮
Example with \( f(x) = 2x + 1 \)
\( f^{-1}(5) = ? \)
Means: \( f(?) = 5 \)
\( 2x + 1 = 5 \) → \( x = 2 \)
✅ \( \boxed{2} \)
Means: \( f(?) = 5 \)
\( 2x + 1 = 5 \) → \( x = 2 \)
✅ \( \boxed{2} \)
\( f^{-1}(x) = 3 \)
Means: \( f(3) = x \)
\( 2(3) + 1 = 7 \)
✅ \( \boxed{7} \)
Means: \( f(3) = x \)
\( 2(3) + 1 = 7 \)
✅ \( \boxed{7} \)
\( f^{-1}(7) = x \)
Means: \( f(x) = 7 \)
\( 2x + 1 = 7 \) → \( x = 3 \)
✅ \( \boxed{3} \)
Means: \( f(x) = 7 \)
\( 2x + 1 = 7 \) → \( x = 3 \)
✅ \( \boxed{3} \)
🎯
Quick summary
- \( f^{-1}(y) = x \) ⇔ \( f(x) = y \)
- To find \( x \) when \( f^{-1}(k) = x \): set \( f(x) = k \) and solve.
- To find \( f^{-1}(k) \): solve \( f(x) = k \) for \( x \).
- No need to find \( f^{-1}(x) \) — just use the meaning.
🔁 One rule: \( f^{-1}(y) = x \) means \( f(x) = y \).
That’s all you need for inverse function questions.
🪞
Even & odd functions
symmetry rules| Type | Rule | Graph symmetry | Example |
|---|---|---|---|
| Even | \(f(-x)=f(x)\) | mirror over y‑axis | \(x^2,\; \cos x\) |
| Odd | \(f(-x)=-f(x)\) | 180° rotation about origin | \(x^3,\; \sin x\) |
| Neither | doesn't satisfy either | no special symmetry | \(x^2+x\) |
✨ Even ⇒ \(f(x) = f(-x)\) | Odd ⇒ \(f(-x) = -f(x)\)
🎯
Domain & Range
allowed inputs → possible outputs- Domain = set of all possible \(x\)-values (no division by zero, no square root of negative).
- Range = all possible \(y\)-values that the function produces.
- For inverses: \(\text{Domain}(f^{-1}) = \text{Range}(f)\) and \(\text{Range}(f^{-1}) = \text{Domain}(f)\).
📌 Example 1: \(f(x)=\frac{1}{x-2}\)
Domain: \(x \neq 2\)
Domain: \(x \neq 2\)
📌 Example 2: \(f(x)=x^2\)
Range: \(y \ge 0\)
Range: \(y \ge 0\)
📈
Graph transformations
shifts, stretches, flips| Change | Rule | Effect (visual) |
|---|---|---|
| Shift up | \(f(x)+a\) | moves graph upward |
| Shift down | \(f(x)-a\) | moves graph downward |
| Shift right | \(f(x-a)\) | slides right |
| Shift left | \(f(x+a)\) | slides left |
| Vertical stretch | \(a\cdot f(x),\; a>1\) | steeper/narrower |
| Reflect in x‑axis | \(-f(x)\) | flip upside down |
| Reflect in y‑axis | \(f(-x)\) | mirror left‑right |
✏️ "inside the bracket: opposite direction" | "outside: up/down stretch"
⚡
Exponential & log rules
same base → add/subtract exponents\(a^m \times a^n = a^{m+n}\)
\(a^m \div a^n = a^{m-n}\)
\((a^m)^n = a^{m \cdot n}\)
\(a^m = a^n \Rightarrow m = n\)
\(\ln(e^x)=x,\quad e^{\ln x}=x\)
\(\log_a(a^x)=x,\quad a^{\log_a x}=x\)
🧠 IGCSE favourite: \(2^{x+1}=8 \;\Rightarrow\; 2^{x+1}=2^3 \;\Rightarrow\; x+1=3 \;\Rightarrow\; x=2\)
⚙️
Differentiation rules (power & more)
gradient of a function| Rule | Formula |
|---|---|
| Power rule | \(\frac{d}{dx}(x^n)=n x^{n-1}\) |
| Sum rule | \(\frac{d}{dx}(f+g)=f'+g'\) |
| Product rule | \(\frac{d}{dx}(fg)=f'g+fg'\) |
| Chain rule | \(\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)\) |
📌 Quick example: \(y = 3x^4 - 2x\) → \(\frac{dy}{dx}=12x^3 - 2\)
📦
Integration rules (reverse of derivative)
area under a curve\(\int x^n dx = \frac{x^{n+1}}{n+1}+C \;(n\neq -1)\)
\(\int e^x dx = e^x + C\)
\(\int \frac{1}{x} dx = \ln|x| + C\)
\(\int k f(x) dx = k\int f(x)dx\)
🧩 Always add “\(+ C\)” for indefinite integrals
📐
Trigonometric rules
sine, cosine & friends\(\sin^2\theta+\cos^2\theta=1\)
\(\sin(-\theta)=-\sin\theta\) (odd)
\(\cos(-\theta)=\cos\theta\) (even)
\(\tan\theta=\frac{\sin\theta}{\cos\theta}\)
💡 For IGCSE: know exact values for 0°,30°,45°,60°,90°
🔑
One‑to‑one (injective) functions
must be one‑to‑one to have an inverseRule: \(f(a)=f(b) \implies a=b\) (different inputs → different outputs).
Horizontal line test: any horizontal line crosses the graph at most once.
✅ \(f(x)=2x+1\) is one‑to‑one ❌ \(f(x)=x^2\) (over all reals) is NOT one‑to‑one
🧠 ⚡ Law of indices (same base) – super important!
If \(a^m = a^n\) and \(a>0,\;a\neq 1\) ⇒ \(m = n\)
Example: \(3^{2x-1}=27\) → \(3^{2x-1}=3^3\) → \(2x-1=3\) → \(x=2\) ✅
🎯 Used with exponential equations and inverse function problems
🌟 These rules are your toolkit for IGCSE Extended Maths (0580). Practice each rule with past paper questions!
📐 Function rules – Examples & exam practice
IGCSE Mathematics | 5 examples + 5 exam‑style questions for each rule
🔄
1. Inverse function rules
✨ 5 examples
Ex 1: \(f(x)=3x-5\). Find \(f^{-1}(x)\).
➜ \(y=3x-5\) → swap \(x,y\): \(x=3y-5\) → \(3y=x+5\) → \(f^{-1}(x)=\frac{x+5}{3}\).
➜ \(y=3x-5\) → swap \(x,y\): \(x=3y-5\) → \(3y=x+5\) → \(f^{-1}(x)=\frac{x+5}{3}\).
Ex 2: \(g(x)=2x+7\). Find \(g^{-1}(11)\).
➜ \(2x+7=11\) → \(2x=4\) → \(x=2\). So \(g^{-1}(11)=2\).
➜ \(2x+7=11\) → \(2x=4\) → \(x=2\). So \(g^{-1}(11)=2\).
Ex 3: \(h(x)=\frac{x}{4}-3\). Find \(h^{-1}(x)\).
➜ \(y=\frac{x}{4}-3\) → swap: \(x=\frac{y}{4}-3\) → \(x+3=\frac{y}{4}\) → \(y=4(x+3)\).
➜ \(y=\frac{x}{4}-3\) → swap: \(x=\frac{y}{4}-3\) → \(x+3=\frac{y}{4}\) → \(y=4(x+3)\).
Ex 4: \(f(x)=5^x\). Find \(f^{-1}(125)\).
➜ \(5^x=125=5^3\) → \(x=3\). Hence \(f^{-1}(125)=3\).
➜ \(5^x=125=5^3\) → \(x=3\). Hence \(f^{-1}(125)=3\).
Ex 5: \(g(x)=x^2\) for \(x\ge0\). Find \(g^{-1}(9)\).
➜ \(x^2=9\) → \(x=3\) (domain restriction).
➜ \(x^2=9\) → \(x=3\) (domain restriction).
📝 5 exam‑style questions
Q1. \(f(x)=4x-9\). Find \(f^{-1}(x)\).
\(y=4x-9\) → swap: \(x=4y-9\) → \(4y=x+9\) → \(f^{-1}(x)=\frac{x+9}{4}\).
Q2. \(h(x)=2x+1\). Calculate \(h^{-1}(-3)\).
\(2x+1=-3\) → \(2x=-4\) → \(x=-2\). So \(h^{-1}(-3)=-2\).
Q3. \(p(x)=\frac{x}{5}+2\). Determine \(p^{-1}(x)\).
\(y=\frac{x}{5}+2\) → swap: \(x=\frac{y}{5}+2\) → \(x-2=\frac{y}{5}\) → \(y=5(x-2)\).
Q4. \(q(x)=3^x\). Evaluate \(q^{-1}(81)\).
\(3^x=81=3^4\) → \(x=4\). So \(q^{-1}(81)=4\).
Q5. \(r(x)=x^2+1\) for \(x\ge0\). Find \(r^{-1}(10)\).
\(x^2+1=10\) → \(x^2=9\) → \(x=3\) (positive root).
🧩
2. Composite functions
✨ 5 examples
Ex 1: \(f(x)=x^2\), \(g(x)=x+3\). \((f\circ g)(x)=(x+3)^2\).
Ex 2: \(f(x)=2x\), \(g(x)=x-4\). \((g\circ f)(x)=2x-4\).
Ex 3: \(f(x)=3x+1\), \(g(x)=x^2\). \((f\circ g)(2)=f(4)=13\).
Ex 4: \(f(x)=x+5\), \(g(x)=2x\). \((f\circ g)(x)=2x+5\); \((g\circ f)(x)=2x+10\) → different.
Ex 5: \(f(x)=x-2\), \(g(x)=3x\). \((f\circ g)(4)=f(12)=10\).
📝 5 exam‑style questions
Q1. \(f(x)=4x\), \(g(x)=x+5\). Find \((f\circ g)(x)\).
\(f(g(x))=4(x+5)=4x+20\).
Q2. \(p(x)=x^2-1\), \(q(x)=2x\). Evaluate \((p\circ q)(3)\).
\(q(3)=6\), \(p(6)=36-1=35\).
Q3. \(f(x)=3x-2\), \(g(x)=\frac{x+2}{3}\). Show \((f\circ g)(x)=x\).
\(f(g(x))=3(\frac{x+2}{3})-2 = x+2-2=x\).
Q4. \(F(x)=x+4\), \(G(x)=x^2\). Find \((G\circ F)(-1)\).
\(F(-1)=3\), \(G(3)=9\).
Q5. \(f(x)=2x+3\), \(g(x)=x-1\). Solve \((f\circ g)(x)=17\).
\(f(g(x))=2(x-1)+3=2x+1\). Set \(2x+1=17\) → \(x=8\).
🪞
3. Even & odd functions
✨ 5 examples
Ex 1: \(f(x)=x^4\) → \(f(-x)=x^4=f(x)\) ⇒ even.
Ex 2: \(g(x)=x^3\) → \(g(-x)=-x^3=-g(x)\) ⇒ odd.
Ex 3: \(h(x)=x^2+x\) → \(h(-x)=x^2-x\) ⇒ neither.
Ex 4: \(p(x)=\cos x\) → even.
Ex 5: \(q(x)=\sin x\) → odd.
📝 5 exam‑style questions
Q1. \(f(x)=5x^2-4\) : even/odd/neither?
\(f(-x)=5x^2-4=f(x)\) → even.
Q2. \(g(x)=2x^3-x\). Is it odd?
\(g(-x)=-2x^3+x = -(2x^3-x)=-g(x)\) → odd.
Q3. \(h(x)=x^2+2x\). Classify.
\(h(-x)=x^2-2x\) ⇒ neither.
Q4. If \(f\) is even and \(f(3)=7\), find \(f(-3)\).
Even ⇒ \(f(-3)=f(3)=7\).
Q5. If \(g\) is odd and \(g(4)=-6\), find \(g(-4)\).
Odd ⇒ \(g(-4)=-g(4)=6\).
🎯
4. Domain & range
✨ 5 examples
Ex 1: \(f(x)=\frac{1}{x-3}\) → domain \(x\neq3\).
Ex 2: \(f(x)=\sqrt{x-5}\) → domain \(x\ge5\), range \(y\ge0\).
Ex 3: \(f(x)=x^2+2\) → domain all reals, range \(y\ge2\).
Ex 4: \(f(x)=3x-1\) for \(-2\le x\le4\) → range \(-7\le y\le11\).
Ex 5: \(f(x)=5^x\) → domain all reals, range \(y>0\).
📝 5 exam‑style questions
Q1. Domain of \(f(x)=\frac{2x}{x+4}\).
\(x\neq -4\).
Q2. Range of \(g(x)=x^2-9\) (all real \(x\)).
Minimum -9 → \(y\ge -9\).
Q3. \(h(x)=\sqrt{2x-6}\): domain & range.
Domain \(x\ge3\), range \(y\ge0\).
Q4. \(f(x)=4-x^2\). State range.
Maximum 4 → \(y\le4\).
Q5. If \(f^{-1}(x)=\frac{x-1}{2}\), what is the range of \(f\)?
Range of \(f\) = domain of \(f^{-1}\) = all reals.
⚡
5. Exponential & log (same base)
✨ 5 examples
Ex 1: \(2^x=32\) → \(2^x=2^5\) ⇒ \(x=5\).
Ex 2: \(3^{2x-1}=27\) → \(3^{2x-1}=3^3\) ⇒ \(2x-1=3\) ⇒ \(x=2\).
Ex 3: \(5^{x^2}=5^9\) ⇒ \(x^2=9\) ⇒ \(x=\pm3\).
Ex 4: \(\log_2 x = 5\) ⇒ \(x=2^5=32\).
Ex 5: \(e^{3x}=e^{12}\) ⇒ \(3x=12\) ⇒ \(x=4\).
📝 5 exam‑style questions
Q1. Solve \(4^x=64\).
\(4^x=4^3\) ⇒ \(x=3\).
Q2. \(7^{2x+1}=343\).
\(343=7^3\) ⇒ \(2x+1=3\) ⇒ \(x=1\).
Q3. \(9^x=27\).
\((3^2)^x=3^3\) ⇒ \(3^{2x}=3^3\) ⇒ \(2x=3\) ⇒ \(x=1.5\).
Q4. \(\log_5 x = 3\).
\(x=5^3=125\).
Q5. \(e^{x-2}=e^{5}\).
\(x-2=5\) ⇒ \(x=7\).
📈
6. Graph transformations
✨ 5 examples
Ex 1: \(f(x)=x^2\) → \(f(x)+3\) shifts up by 3.
Ex 2: \(f(x-2)\) shifts right 2 units.
Ex 3: \(2f(x)\) vertical stretch factor 2.
Ex 4: \(-f(x)\) reflection in x‑axis.
Ex 5: \(f(3x)\) horizontal compression by factor 1/3.
📝 5 exam‑style questions
Q1. \(y=x^2\) shifted left 3 units and up 4. New equation?
\(y=(x+3)^2+4\).
Q2. How is \(g(x)=2f(x+1)\) obtained from \(f(x)\)?
shift left 1, then vertical stretch ×2.
Q3. \(y=f(x)\) reflected in x‑axis → equation?
\(y=-f(x)\).
Q4. If \(h(x)=\sqrt{x-4}\), what transformation of \(\sqrt{x}\)?
shift right by 4.
Q5. \(y=f(2x)\) represents a …?
horizontal compression by factor ½.
🧠 IGCSE Extended Maths (0580) | Click "Show solution" to reveal answers. Use these for revision.
📐 Functions: Inverse & Composite
Two rules · 20 examples · 20 practice questions · no inequality limits
⭐
The only two rules you need
🔁 Inverse: \( f^{-1}(y) = x \) ⇔ \( f(x) = y \)
🧩 Composite: \( (f \circ g)(x) = f(g(x)) \) (do \( g \) first, then \( f \))
🧩 Composite: \( (f \circ g)(x) = f(g(x)) \) (do \( g \) first, then \( f \))
✅ No finding inverses. No inequality limits. Just apply the rules.
📘
20 mixed examples
🔁 Inverse (1–10)
1. \( f(x)=2x+3 \), find \( x \) if \( f^{-1}(x)=4 \)
\( f(4)=11 \) ⇒ \( x=11 \)
\( f(4)=11 \) ⇒ \( x=11 \)
2. \( f(x)=5x-2 \), find \( x \) if \( f^{-1}(x)=3 \)
\( f(3)=13 \) ⇒ \( x=13 \)
\( f(3)=13 \) ⇒ \( x=13 \)
3. \( f(x)=x^2 \), find \( x \) if \( f^{-1}(x)=6 \)
\( f(6)=36 \) ⇒ \( x=36 \)
\( f(6)=36 \) ⇒ \( x=36 \)
4. \( f(x)=x^2+7 \), find \( x \) if \( f^{-1}(x)=4 \)
\( f(4)=23 \) ⇒ \( x=23 \)
\( f(4)=23 \) ⇒ \( x=23 \)
5. \( f(x)=2^x \), find \( x \) if \( f^{-1}(x)=5 \)
\( f(5)=32 \) ⇒ \( x=32 \)
\( f(5)=32 \) ⇒ \( x=32 \)
6. \( f(x)=3^x \), find \( x \) if \( f^{-1}(x)=4 \)
\( f(4)=81 \) ⇒ \( x=81 \)
\( f(4)=81 \) ⇒ \( x=81 \)
7. \( f(x)=4^x+2 \), find \( x \) if \( f^{-1}(x)=3 \)
\( f(3)=66 \) ⇒ \( x=66 \)
\( f(3)=66 \) ⇒ \( x=66 \)
8. \( f(x)=3x-10 \), find \( x \) if \( f^{-1}(x)=-2 \)
\( f(-2)=-16 \) ⇒ \( x=-16 \)
\( f(-2)=-16 \) ⇒ \( x=-16 \)
9. \( f(x)=x^2+12 \), find \( x \) if \( f^{-1}(x)=5 \)
\( f(5)=37 \) ⇒ \( x=37 \)
\( f(5)=37 \) ⇒ \( x=37 \)
10. \( f(x)=5^x \), find \( x \) if \( f^{-1}(x)=2 \)
\( f(2)=25 \) ⇒ \( x=25 \)
\( f(2)=25 \) ⇒ \( x=25 \)
🧩 Composite (11–20)
11. \( f(x)=2x, g(x)=x+3\), find \( f(g(4)) \)
\( g(4)=7, f(7)=14 \)
\( g(4)=7, f(7)=14 \)
12. \( f(x)=x^2, g(x)=x-2\), find \( g(f(3)) \)
\( f(3)=9, g(9)=7 \)
\( f(3)=9, g(9)=7 \)
13. \( f(x)=3x+1, g(x)=x^2\), find \( f(g(2)) \)
\( g(2)=4, f(4)=13 \)
\( g(2)=4, f(4)=13 \)
14. \( f(x)=x+5, g(x)=2x\), find \( g(f(1)) \)
\( f(1)=6, g(6)=12 \)
\( f(1)=6, g(6)=12 \)
15. \( f(x)=4x-3, g(x)=x+2\), find \( f(g(5)) \)
\( g(5)=7, f(7)=25 \)
\( g(5)=7, f(7)=25 \)
16. \( f(x)=2^x, g(x)=x-1\), find \( f(g(3)) \)
\( g(3)=2, f(2)=4 \)
\( g(3)=2, f(2)=4 \)
17. \( f(x)=x^2+1, g(x)=3x\), find \( g(f(2)) \)
\( f(2)=5, g(5)=15 \)
\( f(2)=5, g(5)=15 \)
18. \( f(x)=2x-5, g(x)=x^2\), find \( f(g(4)) \)
\( g(4)=16, f(16)=27 \)
\( g(4)=16, f(16)=27 \)
19. \( f(x)=5x, g(x)=3^x\), find \( f(g(2)) \)
\( g(2)=9, f(9)=45 \)
\( g(2)=9, f(9)=45 \)
20. \( f(x)=2^x+1, g(x)=x+1\), find \( f(g(2)) \)
\( g(2)=3, f(3)=9 \)
\( g(2)=3, f(3)=9 \)
✍️
20 practice questions
🔁 Inverse (1–10)
1. \( f(x)=4x+1 \), find \( x \) if \( f^{-1}(x)=5 \)
2. \( f(x)=7x-3 \), find \( x \) if \( f^{-1}(x)=2 \)
3. \( f(x)=x^2 \), find \( x \) if \( f^{-1}(x)=9 \)
4. \( f(x)=x^2+4 \), find \( x \) if \( f^{-1}(x)=6 \)
5. \( f(x)=3^x \), find \( x \) if \( f^{-1}(x)=4 \)
6. \( f(x)=6^x \), find \( x \) if \( f^{-1}(x)=2 \)
7. \( f(x)=2^x+5 \), find \( x \) if \( f^{-1}(x)=3 \)
8. \( f(x)=5x-9 \), find \( x \) if \( f^{-1}(x)=-1 \)
9. \( f(x)=x^2+11 \), find \( x \) if \( f^{-1}(x)=7 \)
10. \( f(x)=4^x-2 \), find \( x \) if \( f^{-1}(x)=2 \)
🧩 Composite (11–20)
11. \( f(x)=3x, g(x)=x+4 \), find \( f(g(2)) \)
12. \( f(x)=x^2, g(x)=x-1 \), find \( g(f(4)) \)
13. \( f(x)=2x+3, g(x)=x^2 \), find \( f(g(3)) \)
14. \( f(x)=x+6, g(x)=4x \), find \( g(f(1)) \)
15. \( f(x)=5x-2, g(x)=x+7 \), find \( f(g(2)) \)
16. \( f(x)=2^x, g(x)=x-3 \), find \( f(g(5)) \)
17. \( f(x)=x^2+2, g(x)=2x \), find \( g(f(3)) \)
18. \( f(x)=3x-4, g(x)=x^2 \), find \( f(g(5)) \)
19. \( f(x)=4x, g(x)=2^x \), find \( f(g(3)) \)
20. \( f(x)=3^x+2, g(x)=x+2 \), find \( f(g(1)) \)
✅
Answers
1. 21
2. 11
3. 81
4. 40
5. 81
6. 36
7. 13
8. -14
9. 60
10. 14
11. 18
12. 15
13. 21
14. 28
15. 43
16. 4
17. 22
18. 71
19. 32
20. 29
🔁 Rule 1: \( f^{-1}(y)=x \) ⇔ \( f(x)=y \) | 🧩 Rule 2: \( (f \circ g)(x)=f(g(x)) \) (do \( g \) first, then \( f \))
📐 Domain & Range – Learner Guide
How to easily identify and state the domain and range of any function
🎯
What are domain and range?
🔵 Domain
All possible input values (\(x\) values).
👉 "What \(x\) are allowed?"
All possible input values (\(x\) values).
👉 "What \(x\) are allowed?"
🟠 Range
All possible output values (\(y\) values).
👉 "What \(y\) come out?"
All possible output values (\(y\) values).
👉 "What \(y\) come out?"
💡 Remember: Domain = Door (x enters) | Range = Roof (y comes out)
🔍
Step‑by‑step method
📌 1. Identify restrictions
- Denominator ≠ 0 → exclude values that make denominator zero
- Square root: inside must be ≥ 0
- Logarithm: inside must be > 0
- If no restrictions → domain = all real numbers (\(\mathbb{R}\))
📌 2. Find the range
- Think about outputs: squares are ≥0, exponentials >0, etc.
- For quadratics: find vertex, check direction (opens up/down)
- For rational functions: sketch or solve \(y = f(x)\) for \(x\)
⚡
Common functions – domain & range at a glance
| Function | Domain | Range |
|---|---|---|
| \(f(x)=mx+c\) | \(\mathbb{R}\) | \(\mathbb{R}\) |
| \(f(x)=x^2\) | \(\mathbb{R}\) | \(y \ge 0\) |
| \(f(x)=x^2 + k\) | \(\mathbb{R}\) | \(y \ge k\) |
| \(f(x)= -x^2 + k\) | \(\mathbb{R}\) | \(y \le k\) |
| \(f(x)=\frac{1}{x}\) | \(x \neq 0\) | \(y \neq 0\) |
| \(f(x)=\frac{1}{x-a}\) | \(x \neq a\) | \(y \neq 0\) |
| \(f(x)=\sqrt{x}\) | \(x \ge 0\) | \(y \ge 0\) |
| \(f(x)=\sqrt{x-h}\) | \(x \ge h\) | \(y \ge 0\) |
| \(f(x)=a^x\) \((a>0)\) | \(\mathbb{R}\) | \(y > 0\) |
| \(f(x)=a^x + k\) | \(\mathbb{R}\) | \(y > k\) |
| \(f(x)=\log x\) | \(x > 0\) | \(\mathbb{R}\) |
📘
Worked examples (step‑by‑step)
📐 Example 1 – Linear
\(f(x)=3x-2\)
✅ Domain: all real numbers \(\mathbb{R}\) or \((-\infty,\infty)\)
✅ Range: all real numbers \(\mathbb{R}\) or \((-\infty,\infty)\)
\(f(x)=3x-2\)
✅ Domain: all real numbers \(\mathbb{R}\) or \((-\infty,\infty)\)
✅ Range: all real numbers \(\mathbb{R}\) or \((-\infty,\infty)\)
📐 Example 2 – Quadratic
\(f(x)=x^2-4x+5\)
▶ Domain: all real numbers
▶ Vertex: \(x=2\) → \(y=1\) (opens up)
✅ Range: \(y \ge 1\) or \([1,\infty)\)
\(f(x)=x^2-4x+5\)
▶ Domain: all real numbers
▶ Vertex: \(x=2\) → \(y=1\) (opens up)
✅ Range: \(y \ge 1\) or \([1,\infty)\)
📐 Example 3 – Reciprocal
\(f(x)=\frac{2}{x-1}\)
▶ Denominator zero when \(x=1\)
✅ Domain: \(x \neq 1\) → \((-\infty,1)\cup(1,\infty)\)
✅ Range: \(y \neq 0\) → \((-\infty,0)\cup(0,\infty)\)
\(f(x)=\frac{2}{x-1}\)
▶ Denominator zero when \(x=1\)
✅ Domain: \(x \neq 1\) → \((-\infty,1)\cup(1,\infty)\)
✅ Range: \(y \neq 0\) → \((-\infty,0)\cup(0,\infty)\)
📐 Example 4 – Square root
\(f(x)=\sqrt{2x-6}\)
▶ Inside ≥ 0 → \(2x-6 \ge 0 \Rightarrow x \ge 3\)
✅ Domain: \([3,\infty)\)
✅ Range: \([0,\infty)\)
\(f(x)=\sqrt{2x-6}\)
▶ Inside ≥ 0 → \(2x-6 \ge 0 \Rightarrow x \ge 3\)
✅ Domain: \([3,\infty)\)
✅ Range: \([0,\infty)\)
📐 Example 5 – Exponential
\(f(x)=3^x - 2\)
✅ Domain: all real numbers
▶ \(3^x > 0\) → subtract 2 → \(y > -2\)
✅ Range: \((-2,\infty)\)
\(f(x)=3^x - 2\)
✅ Domain: all real numbers
▶ \(3^x > 0\) → subtract 2 → \(y > -2\)
✅ Range: \((-2,\infty)\)
✍️
Practice questions
1. \(f(x)=\frac{3}{x+2}\)
📍 Domain? Range?
📍 Domain? Range?
2. \(g(x)=\sqrt{x-4}\)
📍 Domain? Range?
📍 Domain? Range?
3. \(h(x)=2^x + 5\)
📍 Domain? Range?
📍 Domain? Range?
4. \(p(x)=x^2-6x+10\)
📍 Domain? Range?
📍 Domain? Range?
5. \(q(x)=\frac{1}{x^2}\)
📍 Domain? Range?
📍 Domain? Range?
✅ Click to see answers
1. Domain: \(x \neq -2\) Range: \(y \neq 0\)
2. Domain: \(x \ge 4\) Range: \(y \ge 0\)
3. Domain: all reals Range: \(y > 5\)
4. Domain: all reals Range: \(y \ge 1\) (vertex at \(x=3, y=1\))
5. Domain: \(x \neq 0\) Range: \(y > 0\)
📝
Notation & quick checklist
| Notation | Meaning | Example |
|---|---|---|
| \((a,b)\) | between \(a\) and \(b\), not including \(a,b\) | \((2,5)\) |
| \([a,b]\) | between \(a\) and \(b\), including \(a,b\) | \([2,5]\) |
| \((a,\infty)\) | greater than \(a\) | \((2,\infty)\) |
| \((-\infty,a)\) | less than \(a\) | \((-\infty,5)\) |
| \(\mathbb{R}\) | all real numbers | \(\mathbb{R}\) |
| \(x \neq a\) | all real numbers except \(a\) | \(x \neq 3\) |
✅ Quick checklist – Domain
☐ Denominator ≠ 0 ☐ Square root: inside ≥ 0 ☐ Logarithm: inside > 0
☐ If none → all real numbers
☐ Denominator ≠ 0 ☐ Square root: inside ≥ 0 ☐ Logarithm: inside > 0
☐ If none → all real numbers
✅ Quick checklist – Range
☐ Sketch or think about possible outputs ☐ Quadratics: vertex + direction
☐ Exponentials: base>0 → y>0 (plus shift) ☐ Reciprocals: y ≠ 0
☐ Sketch or think about possible outputs ☐ Quadratics: vertex + direction
☐ Exponentials: base>0 → y>0 (plus shift) ☐ Reciprocals: y ≠ 0
🧠 IGCSE Extended Maths (0580) | Domain = allowed \(x\) values | Range = possible \(y\) values