# Functions

> CIE IGCSE Mathematics · 0580 2025-2027
> Source: https://www.owlsprep.com/study/cie-0580-u2-functions/

This guide covers all Extended-only functions content for CIE IGCSE Maths 0580: function notation, mapping diagrams, domain/range, inverse functions, and composite functions, tailored for Paper 2 and 4 candidates.

**Prerequisites:** [Algebraic rearrangement](https://www.owlsprep.com/study/cie-0580-u2-algebraic-manipulation/); [Linear and rational expression basics](https://www.owlsprep.com/study/cie-0580-u2-algebraic-expressions/)

## Learning objectives

- Use function notation f(x) and interpret mapping diagrams
- Calculate values of a function for given inputs
- Identify the domain and range of basic functions
- Find the inverse function f⁻¹(x) for linear and simple rational functions
- Form and evaluate composite functions gf(x)

## Function Notation, Mapping Diagrams and Evaluating Functions

**Function** — A rule that maps each input value to exactly one output value

*Notation:* f(x)

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 $f(x) = 3x - 7$, 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:

   $$f(-2) = 3(-2) -7 = -6 -7 = -13$$

**Check your understanding**

1. If $g(x) = 2x^2 + 1$, what is $g(3)$?

   *Why:* 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.

## Domain and Range of a Function

**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 $f(x) = 2x + 3$ 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:

   $$Range: 1 \leq f(x) \leq 11$$

> **warning**
>
> Do not mix up domain and range: domain = inputs (x-values), range = outputs (f(x) values). If asked for range, always write it in terms of f(x) or y, not x.

## Finding Inverse Functions f⁻¹(x)

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

*Notation:* f^{-1}(x)

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 $f(x) = \frac{2x + 5}{3}$.

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.

## Composite Functions gf(x)

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

*Notation:* gf(x)

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

**Worked example:** Given $f(x) = 4x - 1$ and $g(x) = x^2 + 2$, find $gf(2)$ and $gf(x)$ 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:

   $$gf(x) = 16x^2 - 8x +1 +2 = 16x^2 -8x +3$$

> **tip**
>
> Write $gf(x)$ as $g(f(x))$ to remind yourself that f is the inner function to apply first.

## Common pitfalls

- **Wrong:** Calculating composite function $gf(x)$ by applying g first then f
  - Why it fails: The order of composite functions is right to left: the function closest to the input is applied first
  - Correct: Always treat the inner function (the one inside the brackets) as the first to apply: for $gf(x)$, compute f(x) first, then substitute into g(x)
- **Wrong:** Writing the range of a function in terms of x instead of f(x) or y
  - Why it fails: Range refers to output values, not input x-values
  - Correct: Label range using f(x) e.g. $2 \leq f(x) \leq 10$, not $2 \leq x \leq10$
- **Wrong:** Forgetting to swap x and y when calculating the inverse function
  - Why it fails: The inverse function is expressed in terms of x as the input, same as the original function
  - Correct: After rearranging to make x the subject, swap x and y before writing the final $f^{-1}(x)$ expression
- **Wrong:** Rejecting a mapping where multiple inputs point to the same output as not a function
  - Why it fails: A function only requires each input to have exactly one output; multiple inputs can share the same output
  - Correct: Reject mappings only if an input has more than one output, not if outputs are shared
- **Wrong:** Attempting to calculate domain or range for composite functions
  - Why it fails: The 0580 Extended syllabus does not require this skill
  - Correct: Focus only on calculating the value or simplified expression of composite functions as required

## Cheatsheet

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

## 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.

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