Functions

Q: What order do I solve composite function gf(x) in?

Always apply the inner function first: for , substitute into first, then substitute the result into .

CIE IGCSE Mathematics· E2.13· 20 min read

1. Function Notation, Mapping Diagrams and Evaluating Functions★★☆☆☆⏱ 5 min

📘 Definition

Function

f(x)

A rule that maps each input value to exactly one output value

Functions are written using notation like $f (x)$ , $g (x)$ or $h (x)$ , where the letter inside the brackets is the input variable. A mapping diagram shows inputs on the left, arrows pointing to their corresponding outputs on the right, with each input having exactly one arrow leaving it.

📐 Worked Example

Given $f (x) = 3 x - 7$ , find the value of $f (4)$ and $f (- 2)$ .

1
To find f(4), substitute x = 4 into the function:
$f(4) = 3(4) - 7 = 12 - 7 = 5$
2
To find f(-2), substitute x = -2 into the function:
$f(-2) = 3(-2) -7 = -6 -7 = -13$

✓ Quick check

If $g (x) = 2 x^{2} + 1$ , what is $g (3)$ ?
Reveal answer
19 —
Substitute x=3: $2 (3^{2}) + 1 = 18 + 1 = 19$

Exam tip:

Double check substitution of negative values, as sign errors are extremely common in these questions.

2. Domain and Range of a Function★★★☆☆⏱ 6 min

📘 Definition

Domain and Range

The domain is the set of all valid input values for a function. The range is the set of all valid output values produced by the function for the given domain.

For most IGCSE questions, domain values are either explicitly stated, or you exclude values that make the function undefined (e.g. division by zero). The range is found by substituting the domain limits into the function, or identifying minimum/maximum output values.

📐 Worked Example

A function is defined as $f (x) = 2 x + 3$ for $- 1 \leq x \leq 4$ . State the domain and range of f(x).

1
The domain is given explicitly in the question:
$Domain: -1 \leq x \leq 4$
2
Find the minimum output by substituting the smallest domain value x=-1:
$f(-1) = 2(-1) +3 = 1$
3
Find the maximum output by substituting the largest domain value x=4:
$f(4) = 2(4) +3 = 11$
4
Since f(x) is a linear increasing function, all values between 1 and 11 are valid outputs:
$Range: 1 \leq f(x) \leq 11$

3. Finding Inverse Functions f⁻¹(x)★★★☆☆⏱ 5 min

📘 Definition

Inverse Function

f^{-1}(x)

The inverse function $f^{- 1} (x)$ reverses the action of the original function f(x). If $f (a) = b$ , then $f^{- 1} (b) = a$ .

To find the inverse of a function, follow these 3 steps: 1. Replace $f (x)$ with $y$ , 2. Rearrange the equation to make $x$ the subject, 3. Replace $x$ with $f^{- 1} (x)$ and $y$ with $x$ .

📐 Worked Example

Find the inverse of the function $f (x) = \frac{2 x + 5}{3}$ .

1
Step 1: Replace f(x) with y:
$y = \frac{2x +5}{3}$
2
Step 2: Rearrange to make x the subject: multiply both sides by 3:
$3y = 2x + 5$
3
Subtract 5 from both sides:
$3y -5 = 2x$
4
Divide both sides by 2:
$x = \frac{3y -5}{2}$
5
Step 3: Swap x and y, replace y with f⁻¹(x):
$f^{-1}(x) = \frac{3x -5}{2}$

Exam tip:

Check if your inverse is correct by substituting a value into f(x), then substituting that result into f⁻¹(x) to get back your original input.

4. Composite Functions gf(x)★★★★☆⏱ 4 min

📘 Definition

Composite Function

gf(x)

A composite function is made by applying one function to the result of another function. $g f (x)$ means 'apply f first, then apply g to the output of f'.

Never reverse the order of composite functions: $g f (x)$ is almost always different from $f g (x)$ . To evaluate a composite function, always start with the inner function first.

📐 Worked Example

Given $f (x) = 4 x - 1$ and $g (x) = x^{2} + 2$ , find $g f (2)$ and $g f (x)$ as an expression in x.

1
First find gf(2): calculate f(2) first:
$f(2) = 4(2) -1 = 8 -1 =7$
2
Substitute the result into g(x):
$g(7) = 7^2 + 2 = 49 + 2 = 51$
3
Now find the general expression for gf(x): substitute f(x) into g(x) wherever there is an x:
$gf(x) = g(f(x)) = (4x -1)^2 + 2$
4
Expand and simplify:
$gf(x) = 16x^2 - 8x +1 +2 = 16x^2 -8x +3$

5. Common Pitfalls

Wrong move:

Calculating composite function $g f (x)$ by applying g first then f

Why:

The order of composite functions is right to left: the function closest to the input is applied first

Correct move:

Always treat the inner function (the one inside the brackets) as the first to apply: for $g f (x)$ , compute f(x) first, then substitute into g(x)

Wrong move:

Writing the range of a function in terms of x instead of f(x) or y

Why:

Range refers to output values, not input x-values

Correct move:

Label range using f(x) e.g. $2 \leq f (x) \leq 10$ , not $2 \leq x \leq 10$

Wrong move:

Forgetting to swap x and y when calculating the inverse function

Why:

The inverse function is expressed in terms of x as the input, same as the original function

Correct move:

After rearranging to make x the subject, swap x and y before writing the final $f^{- 1} (x)$ expression

Wrong move:

Rejecting a mapping where multiple inputs point to the same output as not a function

Why:

A function only requires each input to have exactly one output; multiple inputs can share the same output

Correct move:

Reject mappings only if an input has more than one output, not if outputs are shared

Wrong move:

Attempting to calculate domain or range for composite functions

Why:

The 0580 Extended syllabus does not require this skill

Correct move:

Focus only on calculating the value or simplified expression of composite functions as required

6. Quick Reference Cheatsheet

Concept	Rule	Example
Function notation	$f (x)$ = value of function f at input x	$f (x) = 2 x + 1$ , $f (3) = 7$
Domain	Set of valid input x values	For $f (x) = 1/ x$ , domain $x \neq = 0$
Range	Set of valid output f(x) values	For $f (x) = x^{2}$ , range $f (x) \geq 0$
Inverse $f^{- 1} (x)$	Replace f(x) with y, 2. Rearrange for x, 3. Swap x and y	Inverse of $f (x) = 3 x - 2$ is $f^{- 1} (x) = (x + 2) /3$
Composite $g f (x)$	Apply f first, then substitute result into g	$g f (x) = g (f (x))$

7. Frequently Asked

Do I need to find domain/range for composite functions in 0580 Extended?

No, the 2025-2027 0580 syllabus does not require you to calculate domain or range of composite functions, only of basic single functions.

What order do I solve composite function gf(x) in?

Always apply the inner function first: for $g f (x)$ , substitute $x$ into $f (x)$ first, then substitute the result into $g (x)$ .

Going deeper

What's Next

Now that you have mastered functions for CIE IGCSE 0580 Extended, you can apply this knowledge to solve more complex algebra and graph problems, including questions that combine functions with quadratic or linear graphs. Practice regularly with past paper questions to build speed and accuracy, paying close attention to order of operations for composite functions and common sign errors when calculating inverses. This foundation will also support you if you progress to A-Level Mathematics in the future.