Functions — CIE 9709 Pure 1 Study Guide

For: Cambridge A-Level Mathematics 9709 candidates sitting Paper 1 (Pure Mathematics 1).

Covers: Topic 2 of the syllabus — function definition, domain & range, one-to-one functions, composite functions, inverse functions, quadratic functions, the modulus function, and graph transformations.

You should already know: Algebraic manipulation, sketching basic curves ($y=x^2$, $y=1/x$, $y=\sqrt{x}$), and solving quadratic equations.

A note on the practice questions: All worked questions in Section 10 are original problems written in the CIE 9709 Pure 1 style — same difficulty, same syllabus coverage, same question structure as past papers, but with new numbers and contexts. They are not reproductions of past Cambridge papers. For actual past papers, please refer to your Cambridge teacher resources or licensed past paper sources.

1. What Is a Function?

A function $f$ from a set $A$ to a set $B$ assigns to every element of $A$ exactly one element of $B$. We write:

$$f:A\to B,x\mapsto f(x)$$

Key idea: a function is a rule. Inputs are unique; outputs are determined by the rule.

Notation you must know:

• $f(x)=2x+3$ — defines a rule.
• $f:x\mapsto 2x+3$ — same rule, "maps to" notation.
• $f(5)=13$ — the value of $f$ at $x=5$.

Caution: $y=\pm \sqrt{x}$ is not a function (one input gives two outputs). $y=+\sqrt{x}$ alone is.

2. Domain and Range

• Domain = the set of allowed inputs $x$.
• Range = the set of all output values $f(x)$ as $x$ varies over the domain.

How to find the range

Method 1 — algebraic (preferred for quadratics): complete the square or use calculus.

Example: $f(x)=x^2-4x+7$ for $x\in \mathbb{R}$.

Complete the square: $f(x)=(x-2{)}^2+3$.

Since $(x-2{)}^2\ge 0$, the minimum is $3$ at $x=2$. So range = $f(x)\ge 3$, i.e. $[3,\infty )$.

Method 2 — graphical: sketch and read the $y$-values.

Restricted domains

Many 9709 problems restrict the domain (e.g. $x\ge 2$) precisely so the function becomes one-to-one (and therefore has an inverse — see §4).

3. One-to-One Functions

A function $f$ is one-to-one (injective) if different inputs give different outputs:

$$x_1\ne x_2⟹f(x_1)\ne f(x_2)$$

Horizontal line test: a function is one-to-one if and only if every horizontal line intersects its graph at most once.

Why it matters: only one-to-one functions have inverses (§4).

Example: $f(x)=x^2$ on $\mathbb{R}$ is not one-to-one (since $f(-2)=f(2)=4$). But $f(x)=x^2$ for $x\ge 0$ is one-to-one.

4. Composite Functions

If $f:A\to B$ and $g:B\to C$, the composite $gf$ (or $g\circ f$) is the function

$$gf(x)=g(f(x))$$

Order matters: $gf$ means "apply $f$ first, then $g$." This is not the same as $fg$.

Example: $f(x)=2x+1$ and $g(x)=x^2$.

$$gf(x)=g(2x+1)=(2x+1{)}^2=4x^2+4x+1$$ $$fg(x)=f(x^2)=2x^2+1$$

Clearly $gf\ne fg$ in general.

Domain rule: for $gf$ to make sense, the range of $f$ must be contained in the domain of $g$.

5. Inverse Functions

If $f$ is one-to-one, its inverse $f^{-1}$ undoes $f$:

$$f^{-1}(f(x))=x\text{and}f(f^{-1}(y))=y$$

How to find $f^{-1}$ algebraically

1. Write $y=f(x)$.
2. Swap $x$ and $y$ (or solve for $x$ in terms of $y$, then rename).
3. The result is $f^{-1}(x)$.

Example: $f(x)=3x-5$.

$y=3x-5\Rightarrow x=\frac{y+5}{3}$, so $f^{-1}(x)=\frac{x+5}{3}$.

Domain and range swap

• Domain of $f^{-1}$ = range of $f$.
• Range of $f^{-1}$ = domain of $f$.

Graphs

The graph of $y=f^{-1}(x)$ is the reflection of $y=f(x)$ in the line $y=x$. Always sketch both with the line $y=x$ — examiners reward clear reflections.

6. Quadratic Functions

A quadratic has the form $f(x)=a{x}^2+bx+c$ with $a\ne 0$.

Completing the square (the 9709 power tool)

$$a{x}^2+bx+c=a{\left(x+\frac{b}{2a}\right)}^2+\left(c-\frac{b^2}{4a}\right)$$

This immediately gives:

• Vertex (turning point): $\left(-\frac{b}{2a},c-\frac{b^2}{4a}\right)$
• Minimum value (if $a>0$) or maximum (if $a<0$): $c-\frac{b^2}{4a}$
• Range: $f(x)\ge c-\frac{b^2}{4a}$ (if $a>0$).
• Line of symmetry: $x=-\frac{b}{2a}$.

Discriminant

$\Delta =b^2-4ac$ controls the roots:

• $\Delta >0$: two distinct real roots.
• $\Delta =0$: one repeated root (curve touches $x$-axis).
• $\Delta <0$: no real roots (curve doesn't cross $x$-axis).

9709 trap: many "find the values of $k$ such that..." questions reduce to a discriminant condition.

7. The Modulus Function

The modulus (absolute value) is

$$|x|=\{\begin{array}{ll}x& \text{if }x\ge 0\\ -x& \text{if }x<0\end{array}$$

Sketching $y=|f(x)|$

1. Sketch $y=f(x)$ as usual.
2. Reflect any part below the $x$-axis up over the $x$-axis.

Solving $|f(x)|=g(x)$ — the squaring method

Square both sides (only when both sides are non-negative):

$$|f(x)|=g(x)⟹(f(x){)}^2=(g(x){)}^2$$

Solve, then check each solution in the original equation (squaring can introduce extraneous roots).

Alternative: split into two cases ($f(x)=g(x)$ and $f(x)=-g(x)$), solve each, then check.

8. Graph Transformations

You must master these four transformations of $y=f(x)$:

Order of combined transformations: in $y=af(bx+c)+d$, apply in the order

1. Horizontal shift by $-c/b$.
2. Horizontal stretch by $1/b$.
3. Vertical stretch by $a$.
4. Vertical shift by $d$.

(Most exam tasks chain only 2–3 of these. Always describe each step explicitly when an exam asks.)

9. Common Pitfalls (and how to avoid them)

1. Confusing $f^{-1}(x)$ with $\frac{1}{f(x)}$. $f^{-1}$ is the inverse function, not the reciprocal. Bad answers often confuse the two.
2. Forgetting the domain restriction makes a function one-to-one. If the question gives $x\ge 2$, that restriction is part of the answer when you state $f^{-1}$'s domain.
3. Swapping domain and range. Domain of $f^{-1}$ = range of $f$. Don't write the domain of $f$.
4. Signed shift in $f(x+a)$. $f(x+3)$ shifts the graph left by 3, not right. Drill this until automatic.
5. Squaring modulus equations without checking solutions. Always substitute back into the original equation.
6. Treating quadratic max/min by calculus when "complete the square" was asked. 9709 examiners reward the requested method; using calculus when the question says "by completing the square" loses marks.

10. Practice Questions (CIE 9709 P1 Style)

The two questions below are original practice problems, not reproductions of past Cambridge papers. They are written to match the difficulty, syllabus coverage, and structure of typical CIE 9709 Paper 1 Functions questions.

Practice Question 1 — Composite & Inverse (CIE 9709 P1 style)

The function $f$ is defined by $f(x)=2x+3$ for $x\in \mathbb{R}$, and $g$ is defined by $g(x)=x^2-4$ for $x\in \mathbb{R}$.

(a) Find an expression for $gf(x)$.

Solution. $gf(x)=g(2x+3)=(2x+3{)}^2-4=4x^2+12x+9-4=\boxed{4x^2+12x+5}$

(b) Find $f^{-1}(x)$ and state its range.

Solution. $y=2x+3\Rightarrow x=\frac{y-3}{2}$, so $f^{-1}(x)=\frac{x-3}{2}$. Range of $f^{-1}$ = domain of $f$ = $\mathbb{R}$.

Practice Question 2 — Quadratic with Restricted Domain

The function $f$ is defined by $f(x)=x^2-6x+13$ for $x\ge 3$.

(a) Express $f(x)$ in the form $(x-a{)}^2+b$.

Solution. $f(x)=(x-3{)}^2+4$.

(b) State the range of $f$.

Solution. Since $x\ge 3$, $(x-3{)}^2\ge 0$, so $f(x)\ge 4$. Range: $f(x)\ge 4$.

(c) Find $f^{-1}(x)$, stating its domain.

Solution. $y=(x-3{)}^2+4\Rightarrow (x-3{)}^2=y-4\Rightarrow x-3=\sqrt{y-4}$ (positive root since $x\ge 3$).

So $x=3+\sqrt{y-4}$, giving $f^{-1}(x)=3+\sqrt{x-4}$.

Domain of $f^{-1}$ = range of $f$ = $x\ge 4$.

11. Quick Reference Cheatsheet

What's Next

• Practice: After working through the two practice questions above, attempt CIE 9709 Paper 1 questions on Functions from your past paper collection (May/June and Oct/Nov sessions, variants 11/12/13). Focus on whether you can identify the technique (e.g. "complete the square", "swap to find inverse") within 30 seconds of reading.
• Related topics: Quadratics (Topic 1), Coordinate Geometry (Topic 3), Trigonometry (Topic 4) — composite and inverse functions often combine with these in multi-part exam questions.

Last updated: 2026-05-06. Compiled by OwlsPrep — a study companion built specifically for IB / AP / A-Level students.