Solve the inequality $|2x + 1| \le |x - 5|$.

M1 for squaring both sides to remove the modulus signs: $(2x+1)^2 \le (x-5)^2$. M1 for expanding the brackets: $4x^2 + 4x + 1 \le x^2 - 10x + 25$. M1 for rearranging to form a quadratic inequality: $3x^2 + 14x - 24 \le 0$. M1 for factorising to find the critical values: $(3x-4)(x+6) \le 0$, yielding critical values $-6$ and $\frac{4}{3}$. A1 for stating the correct closed interval $-6 \le x \le \frac{4}{3}$.

Given that $f(x) = \frac{x+1}{x-2}$, $x \ne 2$, find an expression for $ff(x)$ in its simplest form.

M1 for substituting $f(x)$ into itself: $ff(x) = \frac{\frac{x+1}{x-2} + 1}{\frac{x+1}{x-2} - 2}$. M1 for multiplying the numerator and the denominator by $(x-2)$ to clear the nested fractions. M1 for obtaining the expanded fraction $\frac{(x+1) + (x-2)}{(x+1) - 2(x-2)}$. A1 for fully simplifying to $\frac{2x-1}{5-x}$ (or $\frac{1-2x}{x-5}$).

A Level Math Revision

Edexcel IAL P3 Functions: exam questions and answers, notes and worked examples

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

fg(x) means apply g first, then f; gf(x) means apply f first, then g. They are usually different: with f(x) = 2x + 1 and g(x) = x², fg(x) = 2x² + 1 but gf(x) = (2x + 1)².

Q: Why does a function need to be one-to-one to have an inverse?

If two inputs give the same output, the inverse would have to send that output back to two different places, which a function cannot do. Restricting the domain (for example x ≥ 0) makes it one-to-one.

Q: Is a calculator allowed in IAL P3?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P3 included. Method marks are for the working you write down, so show it.

Q: How long is the IAL P3 exam?

Each IAL maths unit, P3 included, is one written paper of 1 hour 30 minutes worth 75 marks.

Q: How often does functions come up in P3?

In our mapping of the 4 Pearson P3 papers from June 2024 to June 2025, functions was tested in 4 of them (9 questions in all). Any topic can be examined in any series.

Updated October 2026 · By Pete Bromfield

P3 functions covers domain and range, composite functions $f g (x) = f (g (x))$ , inverse functions and the modulus function. Work from the inside out in a composite, and remember that only a one-to-one function has an inverse: the domain of $f^{- 1}$ is the range of $f$ , and its graph is the reflection of $y = f (x)$ in $y = x$ .

P3 · Pure Mathematics 3
Calculator allowed
3 worked examples
35 practice questions
Tested in 4 of the last 4 papers

Practise P3 functions →

Notes
Formulas
Worked examples
Mistakes
FAQ

P3 Functions notes

Domain, range and mappings

The domain is the set of inputs; the range is the set of outputs. A function is one-to-one if each output comes from exactly one input. $f (x) = x^{2} + 3$ , $x \in R$ , is many-to-one and its range is $f (x) \geq 3$ .

Composite functions

$f g (x)$ means apply $g$ first, then $f$ . With $f (x) = 2 x + 1$ and $g (x) = x^{2}$ : $f g (x) = 2 x^{2} + 1$ but $g f (x) = (2 x + 1)^{2}$ , and $f g (3) = 19$ . The order matters.

Inverse functions

Write $y = f (x)$ , make $x$ the subject, then swap the letters. For $f (x) = 3 x - 5$ , $f^{- 1} (x) = \frac{x + 5}{3}$ , and $f f^{- 1} (x) = x$ . The domain of $f^{- 1}$ is the range of $f$ ; restrict a many-to-one function's domain before inverting it.

The modulus function

$| x |$ is the size of $x$ without its sign. $y = | f (x) |$ reflects the parts of $y = f (x)$ below the $x$ -axis upwards; $y = f (| x |)$ copies the part for $x \geq 0$ into $x < 0$ . $| 2 x - 3 | = 5$ gives $x = 4$ or $x = - 1$ , and $| x - 2 | < 3$ gives $- 1 < x < 5$ .

Combined transformations

For $y = a f (x + b) + c$ , deal with the inside of the bracket first (the horizontal translation by $- b$ ), then the stretch by factor $a$ , then the vertical translation by $c$ . Track a few key points, such as turning points and intercepts, through each step.

Key formulas

Functions: $f g (x) = f (g (x))$ ; $f^{- 1}$ reflects $f$ in $y = x$ ; modulus graphs Learn it

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P3 formulas on one page (free PDF) →

P3 Functions worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 5 marks · hard

Solve the inequality $| 2 x + 1 | \leq | x - 5 |$ .

Mark scheme

M1 for squaring both sides to remove the modulus signs: $(2 x + 1)^{2} \leq (x - 5)^{2}$ .
M1 for expanding the brackets: $4 x^{2} + 4 x + 1 \leq x^{2} - 10 x + 25$ .
M1 for rearranging to form a quadratic inequality: $3 x^{2} + 14 x - 24 \leq 0$ .
M1 for factorising to find the critical values: $(3 x - 4) (x + 6) \leq 0$ , yielding critical values $- 6$ and $\frac{4}{3}$ .
A1 for stating the correct closed interval $- 6 \leq x \leq \frac{4}{3}$ .

5 steps

Worked example 2 · 4 marks · hard

Given that $f (x) = \frac{x + 1}{x - 2}$ , $x \neq 2$ , find an expression for $f f (x)$ in its simplest form.

Mark scheme

M1 for substituting $f (x)$ into itself: $f f (x) = \frac{\frac{x + 1}{x - 2} + 1}{\frac{x + 1}{x - 2} - 2}$ .
M1 for multiplying the numerator and the denominator by $(x - 2)$ to clear the nested fractions.
M1 for obtaining the expanded fraction $\frac{(x + 1) + (x - 2)}{(x + 1) - 2 (x - 2)}$ .
A1 for fully simplifying to $\frac{2 x - 1}{5 - x}$ (or $\frac{1 - 2 x}{x - 5}$ ).

4 steps

Worked example 3 · 4 marks · medium

Find the inverse function $f^{- 1} (x)$ for $f (x) = \frac{3 x - 1}{2 x + 5}$ , $x \neq - 2.5$ .

Mark scheme

M1 for setting $y = \frac{3 x - 1}{2 x + 5}$ and multiplying by the denominator: $y (2 x + 5) = 3 x - 1$ .
M1 for expanding and grouping all $x$ terms on one side: $2 x y - 3 x = - 5 y - 1$ .
M1 for factorising out $x$ : $x (2 y - 3) = - 5 y - 1$ .
A1 for isolating $x$ and replacing $y$ with $x$ to write the inverse function: $f^{- 1} (x) = \frac{- 5 x - 1}{2 x - 3}$ or equivalent (e.g. $\frac{5 x + 1}{3 - 2 x}$ ).

4 steps

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Common P3 functions mistakes

Composing in the wrong order. $f g (x)$ applies $g$ first. Writing $g (f (x))$ instead is the most common slip.
The inverse's domain. State the domain of $f^{- 1}$ : it is the range of $f$ , not "all real numbers".
Unchecked modulus solutions. $| x + 1 | = 2 x - 3$ gives $x = 4$ or $x = \frac{2}{3}$ from the two cases, but only $x = 4$ works: $x = \frac{2}{3}$ makes the right-hand side negative.
Confusing |f(x)| with f(|x|). The first reflects in the x-axis what lies below it; the second is symmetrical about the y-axis.

Every P3 mistake from the examiner reports, with the paper it came from →

On your calculator: how to find intersections, zeros, maximum and minimum points. A calculator is allowed in every IAL unit.

P3 Functions FAQ

What is the difference between fg(x) and gf(x)?

fg(x) means apply g first, then f; gf(x) means apply f first, then g. They are usually different: with f(x) = 2x + 1 and g(x) = x², fg(x) = 2x² + 1 but gf(x) = (2x + 1)².

Why does a function need to be one-to-one to have an inverse?

If two inputs give the same output, the inverse would have to send that output back to two different places, which a function cannot do. Restricting the domain (for example x ≥ 0) makes it one-to-one.

Is a calculator allowed in IAL P3?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P3 included. Method marks are for the working you write down, so show it.

How long is the IAL P3 exam?

Each IAL maths unit, P3 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does functions come up in P3?

In our mapping of the 4 Pearson P3 papers from June 2024 to June 2025, functions was tested in 4 of them (9 questions in all). Any topic can be examined in any series.

Next steps

More P3: Numerical Methods · Further Differentiation · Further Integration · Algebraic Methods · Trigonometric Functions · Trigonometric Addition Formulae · Exponentials and Logarithms

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Other units: IAL P1 · IAL P2 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P3 Functions: exam questions and answers, notes and worked examples

Try it first

P3 Functions notes

Domain, range and mappings

Composite functions

Inverse functions

The modulus function

Combined transformations

Key formulas

P3 Functions worked examples

Worked example 1 · 5 marks · hard

Mark scheme

Worked example 2 · 4 marks · hard

Mark scheme

Worked example 3 · 4 marks · medium

Mark scheme

Common P3 functions mistakes

P3 Functions FAQ

What is the difference between fg(x) and gf(x)?

Why does a function need to be one-to-one to have an inverse?

Is a calculator allowed in IAL P3?

How long is the IAL P3 exam?

How often does functions come up in P3?

Next steps

Your P3 revision kit