Functions: the vocabulary and the notation

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A mapping is a rule that links each input to an output.

A function is a special kind of mapping: every input it is allowed to take gives exactly one output.

Almost everything in this unit, and much of the rest of the course, is built on that one idea, so the words that describe functions need to be exact.

Domain and range

The domain of a function is the set of inputs it is allowed to take.

It is part of the function, not an afterthought: $f(x) = x^2$ for $x \in \mathbb{R}$ and $f(x) = x^2$ for $x \geq 0$ are different functions.

The symbol $\mathbb{R}$ means all real numbers, and $x \in \mathbb{R}$ is read " $x$ belongs to the real numbers".

The range (also called the image set) is the set of outputs the function actually produces from its domain.

For $f(x) = x^2$ , $x \in \mathbb{R}$ , every output is zero or positive, and every such value is reached, so the range is $f(x) \geq 0$ .

The range is written in terms of $f(x)$ (or $y$ ), never in terms of $x$ .

Why a mapping is (or is not) a function

To explain in words why a mapping is a function, say that each value in the domain is mapped to exactly one value.

Two things can stop a mapping being a function:

some input has two or more outputs, as with " $x$ maps to $\pm\sqrt{x}$ ", where $9$ maps to both $3$ and $-3$ ;

some input in the stated domain has no output, as with $\dfrac{1}{x}$ on a domain that includes $0$ .

On a graph, a function is a curve that every vertical line in the domain crosses exactly once.

One-one and many-one

A function is one-one if each output comes from exactly one input: different inputs always give different outputs.

$f(x) = 3x - 2$ is one-one.

A function is many-one if at least one output comes from two or more inputs: $f(x) = x^2 + 1$ is many-one because $f(2) = f(-2) = 5$ .

Many-one is still a function.

On a graph, a one-one function is one that every horizontal line meets at most once.

These two words matter because only a one-one function has an inverse function $f^{-1}$ , which reverses $f$ and takes each output back to its input.

A composite function (the composition of functions) is two functions applied one after the other; both are taught fully later in this unit.

The notation you must read and write

Notation

Meaning

$f(x) = 4e^x$

the function $f$ sends $x$ to $4e^x$

$f : x \mapsto 3\lg x$ , for $x > 0$

the same idea in mapping notation, read " $f$ maps $x$ to $3\lg x$ "; the domain is $x > 0$

$f^{-1}(x)$

the inverse function of $f$ applied to $x$

$fg(x)$

$f(g(x))$ : apply $g$ first, then $f$

$f^2(x)$

$f(f(x))$ : apply $f$ twice

Here $\lg x$ means $\log_{10} x$ .

Two of these are easy to misread.

In $fg(x)$ the function written nearest to $x$ acts first.

And $f^2(x)$ means "do $f$ twice"; it is not $\left[f(x)\right]^2$ , and $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$ .

Because $\sin^2 x$ already means $(\sin x)^2$ , the notation $f^2$ is never used with trigonometric functions in this syllabus.

Key terms

Function: a mapping in which every element of the domain is mapped to exactly one element.

Domain: the set of input values for which the function is defined.

Range (image set): the set of output values the function takes over its domain.

One-one function: a function in which each output comes from exactly one input.

Many-one function: a function in which some output comes from more than one input.

Worked example 1: reading function notation (no calculator)

$f(x) = 3e^x$ for $x \in \mathbb{R}$ , and $g : x \mapsto 1 + \lg x$ for $x > 0$ .

Find $g(100)$ , $fg(1000)$ and $f^2(0)$ , giving exact answers.

1. $g(100) = 1 + \lg 100 = 1 + 2 = 3$ , because $10^2 = 100$ .

2. $fg(1000)$ means $f(g(1000))$ . Work from the inside: $g(1000) = 1 + \lg 1000 = 4$ .

3. Then $f(4) = 3e^4$ .

4. $f^2(0)$ means $f(f(0))$ . First $f(0) = 3e^0 = 3$ , then $f(3) = 3e^3$ .

Answer: $g(100) = 3$ , $fg(1000) = 3e^4$ and $f^2(0) = 3e^3$ .

Worked example 2: classifying mappings

For each mapping, state whether it is a function and, if so, whether it is one-one or many-one.

Give a reason each time.

(a) $p(x) = x^2 + 1$ , $x \in \mathbb{R}$ (b) $q(x) = 3x - 2$ , $x \in \mathbb{R}$ (c) $x \mapsto \pm\sqrt{x}$ , $x \geq 0$ (d) $r(x) = \dfrac{1}{x}$ , $x \in \mathbb{R}$

1. (a) Each $x$ gives one value of $x^2 + 1$ , so $p$ is a function. $p(2) = p(-2) = 5$ , so it is many-one.

2. (b) Each $x$ gives one value, and different $x$ give different values of $3x - 2$ , so $q$ is a one-one function.

3. (c) $9$ maps to $3$ and to $-3$ : one input has two outputs, so this is not a function.

4. (d) $0$ is in the stated domain but $\dfrac{1}{0}$ is undefined, so $0$ has no image and $r$ is not a function on this domain. With domain $x \neq 0$ it would be a one-one function.

Answer: (a) many-one function; (b) one-one function; (c) not a function; (d) not a function on $\mathbb{R}$ .

Exam tip

When you explain why a mapping is a function, use the words "each value of $x$ in the domain gives exactly one value of $f(x)$ ".

When you say a function is many-one, back it up with a specific pair of inputs that share an output, such as $p(2) = p(-2)$ .

Common mistakes

Saying a mapping is not a function because two inputs give the same output. That makes it many-one, which is still a function; it fails only when one input has two outputs, or none.

Reading $fg(x)$ as " $f$ first". The function next to $x$ acts first: $fg(x) = f(g(x))$ .

Writing $f^2(x) = \left[f(x)\right]^2$ , or $f^{-1}(x) = \dfrac{1}{f(x)}$ . Both are wrong.

Writing a range in terms of $x$ . The range is a set of outputs, so write it with $f(x)$ or $y$ .

Check your understanding

1. (no calculator) $f(x) = 3 - x^2$ , $x \in \mathbb{R}$ . State whether $f$ is one-one or many-one, giving a reason. [2]

2. (no calculator) $f : x \mapsto 2x + 1$ and $g : x \mapsto x^2$ , both for $x \in \mathbb{R}$ . Find $fg(3)$ and $gf(3)$ . [2]

3. (no calculator) $h(x) = \lg x$ , $x > 0$ . Find the value of $h^2(10^{10})$ . [2]

4. (no calculator) Explain why the mapping $x \mapsto y$ , where $y^2 = x$ and $x \geq 0$ , is not a function. [1]

5. (no calculator) $f(x) = 3e^x$ , $x \in \mathbb{R}$ . Find the exact value of $x$ for which $f(x) = 12$ . [2]

Answers

1. Many-one [1], because, for example, $f(1) = f(-1) = 2$ [1]

2. $fg(3) = f(9) =$ 19 [1]; $gf(3) = g(7) =$ 49 [1]

3. $h(10^{10}) = 10$ [1]; $h(10) = 1$ , so $h^2(10^{10}) =$ 1 [1]

4. A positive $x$ , such as $4$ , gives two values of $y$ ( $2$ and $-2$ ), so one input has two outputs. [1]

5. $3e^x = 12$ , so $e^x = 4$ [1]; $x = \ln 4$ [1]

Functions: the vocabulary and the notation

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