Functions, Domain and Range

Q: Is f²(x) ever interpreted as [f(x)]² in 0606 exams?

No. Per 0606 official specification, f²(x) always means repeated composition f(f(x)) for all question types. If the question wants you to square the output of f(x), it will explicitly write [f(x)]².

Q: What is the difference between fg(x) and f(x)g(x)?

fg(x) is a composite function, meaning you apply g(x) first then substitute the result into f(x). f(x)g(x) is the product of the two functions, meaning you multiply their outputs together. The order of operations is completely different for these two expressions.

CIE IGCSE Additional Mathematics· Syllabus sections 1.1, 1.2, 1.3, 1.5· 45 min read

1. 1. Key Function Terminology & Notation★★☆☆☆⏱ 10 min

📘 Definition

Function

f(x)

A rule that maps every input value from its domain to exactly one output value in its range.

Example:

f(x) = 2x + 3 maps input x = 1 to output f(1) = 5

0606 uses standard function notation with strict rules for special symbols. You will encounter composite functions, repeated composition, and inverse function notation regularly across both papers, so memorizing these rules is critical to avoid easy mark loss.

Cambridge also defines functions using mapping notation: $f : x \mapsto 2 e^{x}$ means exactly the same as $f (x) = 2 e^{x}$ (read it as f maps x to $2 e^{x}$ ). For example, $f : x \mapsto l g x$ is identical to $f (x) = l g x$ , where $l g$ means $lo g_{10}$ . Recognise $f : x \mapsto \dots$ as just another way of writing $f (x) = \dots$ so it does not catch you out in the exam.

📐 Worked Example

Given f(x) = 3x - 2, calculate f(4) and f²(2).

1
Calculate f(4) by substituting x = 4 into the function:
$f(4) = 3(4) - 2 = 12 - 2 = 10$
2
Calculate f²(2) as f(f(2)) (per 0606 notation rules):
$f(2) = 3(2) - 2 = 4$
3
Substitute the result into f(x) again:
$f(4) = 10, \text{ so } f^2(2) = 10$

Exam tip:

Always interpret f²(x) as f(f(x)) in 0606, never square the output of f(x) unless explicitly written as [f(x)]².

2. 2. Domain & Range Calculation★★★☆☆⏱ 15 min

📘 Definition

Domain & Range

Domain = set of all valid input values (x) for which the function is defined. Range = set of all output values (f(x)) produced by the function from its domain (also called the image set).

Common domain restrictions: denominators cannot equal 0, expressions under square roots must be ≥ 0, logarithms only take positive inputs
To find range: identify the minimum and maximum possible output values of the function across its domain, either algebraically or using graph features like vertices

📐 Worked Example

Find the domain and range of $f (x) = x - 2 + 3$ .

1
Find domain: the expression under the square root must be non-negative:
$x - 2 \geq 0 \implies x \geq 2$
2
Find range: the minimum value of $x - 2$ is 0, so the minimum output of f(x) is:
$0 + 3 = 3$
3
Final results: Domain = $x \geq 2$ , Range = $f (x) \geq 3$

📐 Worked Example

Find the domain and range of $f (x) = ln (2 x - 6)$ . Give the domain in the form $x > a$ , where $a$ is as small as possible. (The natural logarithm $ln$ is studied fully in Unit 6; here we only need the rule that its input must be positive.)

1
A logarithm is only defined when its input is strictly positive, so set the argument greater than 0:
$2x - 6 > 0$
2
Solve for the smallest value $x$ can take:
$2x > 6 \implies x > 3$
3
So the domain is $x > 3$ (the least possible value of $a$ is 3). As $x$ decreases towards 3 the argument tends to 0 and $ln$ tends to $- \infty$ ; as $x$ grows the argument grows without bound, so $ln$ takes every real value.
$\text{Domain: } x > 3, \quad \text{Range: } f(x) \in \mathbb{R}$

Exam tip:

Sketch a quick graph of the function if you are unsure of the range, especially for quadratics and root functions. For Paper 2 you may use a scientific calculator, but always show algebraic working for full marks; to check your answer, substitute it back into the function or sketch the graph by hand rather than relying on the calculator.

3. 3. One-One vs Many-One Functions★★★☆☆⏱ 12 min

📘 Definition

One-one & Many-one Functions

One-one function: every unique output corresponds to exactly one unique input. Many-one function: two or more distinct inputs produce the same output.

Example:

The horizontal line test can be used to classify functions: if any horizontal line intersects the graph more than once, the function is many-one.

Only one-one functions have a valid inverse function. Many-one functions cannot have inverses because there is no way to map a shared output back to a single unique input. This is a common short-answer question worth 2-3 marks on 0606 papers.

📐 Worked Example

Explain why $f (x) = x^{2}$ (domain: all real x) is not one-one.

1
Find two distinct input values that produce the same output:
$f(2) = 2^2 = 4 \text{ and } f(-2) = (-2)^2 = 4$
2
Since two different inputs give the same output, the function is many-one, not one-one.

Exam tip:

When asked to explain why a function has no inverse, always state explicitly that it is many-one, and give a concrete example of two inputs giving the same output for full marks.

4. 4. Composite & Inverse Functions★★★★☆⏱ 18 min

📘 Definition

Composite & Inverse Functions

$f g (x), f^{- 1} (x)$

Composite function fg(x) = f(g(x)): apply function g first, then apply f to the result. Inverse function f⁻¹(x) reverses the action of f, and only exists if f is one-one.

Composite function questions often require you to substitute one function into another, and may be combined with domain/range questions or equation-solving. Inverse function questions will often ask you to justify why an inverse does not exist before asking you to calculate an inverse for a restricted domain version of the function.

📐 Worked Example

Given $f (x) = 2 x + 5$ and $g (x) = x^{2}$ , calculate $f g (3)$ and explain why $g (x)$ (domain all real x) has no inverse.

1
Calculate fg(3) = f(g(3)): first find the output of g(3):
$g(3) = 3^2 = 9$
2
Substitute the result into f(x):
$f(9) = 2(9) + 5 = 23 \text{, so } fg(3) = 23$
3
Justify no inverse for g(x): g(x) = x² is many-one, e.g. g(2) = g(-2) = 4, so it is not one-one, therefore no inverse exists.

Exam tip:

Remember the order of composite functions: fg(x) means apply g first, then f. Students often mix up the order and lose easy marks, so rewrite fg(x) as f(g(x)) to remind yourself of the order.

5. Common Pitfalls

Wrong move:

Interpreting f²(x) as [f(x)]² (squared output)

Why:

In other curricula this notation may mean squaring, but 0606 explicitly defines f²(x) as repeated composition.

Correct move:

Always compute f²(x) = f(f(x)) unless the question explicitly writes [f(x)]².

Wrong move:

Calculating composite function fg(x) as f(x) * g(x) (product instead of composition)

Why:

The missing multiplication sign is often confused with a product operation.

Correct move:

Rewrite fg(x) as f(g(x)) first to remind yourself to apply g first, then f, with no multiplication.

Wrong move:

Using the vertex of a quadratic to find range without checking if it lies within the given domain

Why:

Restricted domains may exclude the vertex, so the minimum/maximum value may be at the edge of the domain instead.

Correct move:

Always verify if the vertex of the quadratic falls within the stated domain before using it to calculate range.

Wrong move:

Stating a function has no inverse because it 'fails the vertical line test'

Why:

The vertical line test only checks if a relation is a function, not if it has an inverse.

Correct move:

Explicitly state the function is many-one, and provide a concrete example of two inputs giving the same output to justify no inverse exists.

6. Quick Reference Cheatsheet

Notation/Term	Meaning	Key Exam Rule
f(x)	Function of x	Substitute input value for x to find output
f²(x)	Repeated composition f(f(x))	Never interpret as [f(x)]² for 0606
fg(x)	Composite function f(g(x))	Apply g first, then f to the result
f⁻¹(x)	Inverse function of f	Only exists if f is one-one
Domain	Valid input values	Check for zero denominators, non-negative roots, positive log inputs
Range	Output values (image set)	Use graph or algebraic bounds to find min/max values

7. Frequently Asked

Is f²(x) ever interpreted as [f(x)]² in 0606 exams?

No. Per 0606 official specification, f²(x) always means repeated composition f(f(x)) for all question types. If the question wants you to square the output of f(x), it will explicitly write [f(x)]².

What is the difference between fg(x) and f(x)g(x)?

fg(x) is a composite function, meaning you apply g(x) first then substitute the result into f(x). f(x)g(x) is the product of the two functions, meaning you multiply their outputs together. The order of operations is completely different for these two expressions.

How do I justify that a function has no inverse for exam marks?

You must explicitly state that the function is not one-one (many-one), and provide a concrete example of two distinct input values that produce the same output. Vague statements like 'it fails the line test' will not get full marks.

Going deeper

syllabusCIE 0606 2025-2027 SyllabusRefer to section 1 for full function requirements

What's Next

Now that you have mastered function basics, domain, range, and composite/inverse function rules, you are ready to move on to more advanced function topics in CIE IGCSE Additional Mathematics 0606. Next, you will learn to calculate inverse functions algebraically, solve function equations, and apply function concepts to calculus and trigonometry problems that appear frequently in both Paper 1 (non-calculator) and Paper 2 (calculator) exams. Make sure you practice identifying domain restrictions and justifying inverse existence, as these are common 2-3 mark questions that can be answered quickly if you know the rules.