What's the difference between $f(g(x))$ and $g(f(x))$?

The order of operations is different, which usually leads to a different final function. For $f(g(x))$, you apply the function $g$ first, then apply $f$ to that result. For $g(f(x))$, you apply $f$ first, then apply $g$. Think of it as putting on socks then shoes ($f(g(x))$) versus shoes then socks ($g(f(x))$) – the order matters!

Why do we need to restrict the domain of $y=x^2$ to find its inverse?

The function $f(x)=x^2$ is not one-to-one because different inputs can give the same output (e.g., $f(2)=4$ and $f(-2)=4$). An inverse must map an output back to a *single* unique input. By restricting the domain to, for example, $x \ge 0$, we ensure that every output comes from only one input, which allows an inverse function, $f^{-1}(x) = \sqrt{x}$, to exist.

How do I find the range of a quadratic function quickly?

The best method is to complete the square to write the quadratic $ax^2+bx+c$ in the form $a(x-h)^2+k$. The vertex is at $(h, k)$. If the parabola opens upwards (i.e., $a>0$), the range is $y \ge k$. If it opens downwards (i.e., $a<0$), the range is $y \le k$. Always remember to check if the domain is restricted, as this might alter the range.

Is $f^{-1}(x)$ the same as $\frac{1}{f(x)}$?

No, this is a very important distinction. $f^{-1}(x)$ denotes the inverse function, which reverses the mapping. $\frac{1}{f(x)}$ is the reciprocal of the function's value, sometimes written as $[f(x)]^{-1}$. They are completely different concepts.

Where can I learn Functions for Cambridge 9709?

This free lesson on MarkScheme covers syllabus point 1.2 (Functions) for Cambridge A-Level Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9709/1-2-functions.

Is Functions (1.2) on 9709 free to study?

Yes. The Mathematics 9709 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Functions.

Functions 9709 — Free A-Level Maths

Functions

Welcome to the study of functions. Functions are the fundamental building blocks of calculus and many other areas of mathematics. They describe relationships between variables. In this topic, we'll learn the formal language to describe these relationships, how to combine them, and how to 'undo' them.

BY THE END, YOU CAN…

1Understand and identify the domain and range of a function.
2Form and evaluate composite functions, such as fg(x).
3Determine if a function is one-to-one and find its inverse, f⁻¹(x).
4Understand the graphical relationship between a function and its inverse.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Function Machine

A function is like a mathematical machine that takes an input and produces exactly one specific output. We study how to define the allowed inputs (domain), what outputs are possible (range), and how to chain or reverse these machines.

THINK OF IT LIKE…

Think of a coffee machine. You press the 'Espresso' button (the input, x) and it gives you an espresso (the output, f(x)). Every time you press that same button, you get the same result. The 'domain' is all the buttons on the machine, and the 'range' is all the drinks it can possibly make. An 'inverse function' would be like a machine that takes the espresso and tells you which button was pressed.

1
Function: each input x maps to exactly one output f(x).
2
Domain (inputs) and range (outputs) — restrictions from √(·), ln(·), ÷0.
3
Composite f∘g: apply g first, then f.
4
Inverse f⁻¹ exists only for one-to-one functions; reflection in y = x.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

fg(x) = f(g(x))

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Understanding Functions, Domain, and Range

A function is a rule that assigns to each input value exactly one output value. We often write $y = f(x)$ , where $x$ is the input and $y$ is the output. The set of all permissible input values is called the domain. The set of all resulting output values is called the range.

When determining the domain, we must be careful about two common restrictions: we cannot divide by zero, and we cannot take the square root of a negative number in the real number system. For example, for $f(x) = \frac{1}{x-2}$ , the domain is all real numbers except $x=2$ , written as $x \in \mathbb{R}, x \ne 2$ . For $g(x) = \sqrt{x}$ , the domain is $x \ge 0$ .

Domain: The set of all possible $x$ -values.
Range: The set of all possible $f(x)$ or $y$ -values.
Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.
Finding Range: For quadratics, complete the square to find the vertex. This gives the minimum or maximum value, which defines the range (if the domain is unrestricted).

Composite Functions

A composite function is created when one function is applied to the result of another. The notation $fg(x)$ means we first apply the function $g$ to $x$ , and then apply the function $f$ to the output, $g(x)$ . It's important to work from the inside out. Note that in general, $fg(x)$ is not the same as $gf(x)$ .

fg(x) = f(g(x))

Inverse Functions

An inverse function, denoted $f^{-1}(x)$ , 'reverses' the action of the original function $f(x)$ . If $f(a) = b$ , then $f^{-1}(b) = a$ . A function can only have an inverse if it is one-to-one, meaning every output corresponds to a unique input. We can test for this graphically using the horizontal line test: if any horizontal line cuts the graph more than once, the function is not one-to-one and does not have an inverse over that domain.

Existence: An inverse $f^{-1}$ exists if and only if $f$ is one-to-one.
Algebraic Method: To find $f^{-1}(x)$ , let $y=f(x)$ , swap $x$ and $y$ , then solve for the new $y$ .
Domain and Range Swap: The domain of $f^{-1}$ is the range of $f$ . The range of $f^{-1}$ is the domain of $f$ .
Graphical Property: The graph of $y=f^{-1}(x)$ is a reflection of the graph of $y=f(x)$ in the line $y=x$ .

EXAM TIP

A very common mistake is to correctly find the algebraic expression for $f^{-1}(x)$ but forget to state its domain. The domain of the inverse function is always required unless stated otherwise. Remember: Domain of $f^{-1}$ = Range of $f$ . Always find the range of the original function first!

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A function $f$ is defined by $f(x) = x^2 - 6x + 5$ for $x \in \mathbb{R}$ .

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

(b) Hence, find the range of $f$ .

(c) A function $g$ is defined by $g(x) = x^2 - 6x + 5$ for $x \ge k$ . State the smallest value of $k$ for which $g$ has an inverse.

(d) For this value of $k$ , find an expression for $g^{-1}(x)$ and state its domain.

1
(a) To complete the square: $f(x) = (x^2 - 6x) + 5$ . We halve the coefficient of $x$ , which is -3. So, $f(x) = (x-3)^2 - (-3)^2 + 5 = (x-3)^2 - 9 + 5$ . Therefore, $f(x) = (x-3)^2 - 4$ . [1 mark]

EXAMPLE 2

The functions $f$ and $g$ are defined by $f(x) = 2x+1$ for $x \in \mathbb{R}$ and $g(x) = \frac{12}{x+2}$ for $x \ne -2$ .

(a) Find an expression for $fg(x)$ and simplify your answer.

(b) Find the range of $fg$ .

1
(a) $fg(x) = f(g(x))$ . We substitute the expression for $g(x)$ into $f(x)$ . $fg(x) = 2(g(x)) + 1 = 2\left(\frac{12}{x+2}\right) + 1$ [1 mark] $fg(x) = \frac{24}{x+2} + 1$ . To simplify, we find a common denominator: $fg(x) = \frac{24}{x+2} + \frac{x+2}{x+2} = \frac{24 + x + 2}{x+2} = \frac{x+26}{x+2}$ . [1 mark]

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAFunctions

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 13

What is a function?

TAP OR SPACE TO FLIP

ANSWER

A rule that maps each element in the domain (input set) to exactly one element in the range (output set).

TAP OR SPACE TO FLIP BACK

1 / 13 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Domain: The set of all possible $x$ -values.
✓
Range: The set of all possible $f(x)$ or $y$ -values.
✓
Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.
✓
Finding Range: For quadratics, complete the square to find the vertex. This gives the minimum or maximum value, which defines the range (if the domain is unrestricted).

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[5]

9709/12 · Q2

The curve y = x^2 is transformed to the curve y = 4(x-3)^2 - 8. Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations have been applied.

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do 9709/12 · Q2 on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.1 Quadratics NEXT →1.3 Coordinate geometry

Functions

BY THE END, YOU CAN…

1Understand and identify the domain and range of a function.
2Form and evaluate composite functions, such as fg(x).
3Determine if a function is one-to-one and find its inverse, f⁻¹(x).
4Understand the graphical relationship between a function and its inverse.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Function Machine

THINK OF IT LIKE…

1
Function: each input x maps to exactly one output f(x).
2
Domain (inputs) and range (outputs) — restrictions from √(·), ln(·), ÷0.
3
Composite f∘g: apply g first, then f.
4
Inverse f⁻¹ exists only for one-to-one functions; reflection in y = x.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

fg(x) = f(g(x))

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Understanding Functions, Domain, and Range

Domain: The set of all possible $x$ -values.
Range: The set of all possible $f(x)$ or $y$ -values.
Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.
Finding Range: For quadratics, complete the square to find the vertex. This gives the minimum or maximum value, which defines the range (if the domain is unrestricted).

Composite Functions

fg(x) = f(g(x))

Inverse Functions

Existence: An inverse $f^{-1}$ exists if and only if $f$ is one-to-one.
Algebraic Method: To find $f^{-1}(x)$ , let $y=f(x)$ , swap $x$ and $y$ , then solve for the new $y$ .
Domain and Range Swap: The domain of $f^{-1}$ is the range of $f$ . The range of $f^{-1}$ is the domain of $f$ .
Graphical Property: The graph of $y=f^{-1}(x)$ is a reflection of the graph of $y=f(x)$ in the line $y=x$ .

EXAM TIP

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A function $f$ is defined by $f(x) = x^2 - 6x + 5$ for $x \in \mathbb{R}$ .

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

(b) Hence, find the range of $f$ .

(c) A function $g$ is defined by $g(x) = x^2 - 6x + 5$ for $x \ge k$ . State the smallest value of $k$ for which $g$ has an inverse.

(d) For this value of $k$ , find an expression for $g^{-1}(x)$ and state its domain.

1
(a) To complete the square: $f(x) = (x^2 - 6x) + 5$ . We halve the coefficient of $x$ , which is -3. So, $f(x) = (x-3)^2 - (-3)^2 + 5 = (x-3)^2 - 9 + 5$ . Therefore, $f(x) = (x-3)^2 - 4$ . [1 mark]

EXAMPLE 2

The functions $f$ and $g$ are defined by $f(x) = 2x+1$ for $x \in \mathbb{R}$ and $g(x) = \frac{12}{x+2}$ for $x \ne -2$ .

(a) Find an expression for $fg(x)$ and simplify your answer.

(b) Find the range of $fg$ .

1
(a) $fg(x) = f(g(x))$ . We substitute the expression for $g(x)$ into $f(x)$ . $fg(x) = 2(g(x)) + 1 = 2\left(\frac{12}{x+2}\right) + 1$ [1 mark] $fg(x) = \frac{24}{x+2} + 1$ . To simplify, we find a common denominator: $fg(x) = \frac{24}{x+2} + \frac{x+2}{x+2} = \frac{24 + x + 2}{x+2} = \frac{x+26}{x+2}$ . [1 mark]

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAFunctions

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 13

What is a function?

TAP OR SPACE TO FLIP

ANSWER

A rule that maps each element in the domain (input set) to exactly one element in the range (output set).

TAP OR SPACE TO FLIP BACK

1 / 13 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Domain: The set of all possible $x$ -values.
✓
Range: The set of all possible $f(x)$ or $y$ -values.
✓
Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.
✓
Finding Range: For quadratics, complete the square to find the vertex. This gives the minimum or maximum value, which defines the range (if the domain is unrestricted).

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[5]

9709/12 · Q2

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do 9709/12 · Q2 on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.1 Quadratics NEXT →1.3 Coordinate geometry

Revise Functions for Cambridge 9709 (1.2)

In simple terms

Explore the concept

Key formulas

Full topic notes

Understanding Functions, Domain, and Range

Composite Functions

Inverse Functions

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

9709/12 · Q2

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

Understanding Functions, Domain, and Range

Composite Functions

Inverse Functions

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

9709/12 · Q2

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Functions