3.2 - Functions, domain and range

Learn to read a function as a rule connecting inputs and outputs. You will use function notation, identify forbidden inputs and find the outputs actually reached from a given domain. This lesson is for Higher students.

Mappings and function notation (Higher)

If two different inputs both produce $5$ , does the rule stop being a function? Think of a machine that squares a number, then adds $1$ : both $-2$ and $2$ produce $5$ .

You already know

A set lists each distinct member once. When substituting a negative number into a square, keep brackets: $(-2)^2=4$ .

The important question is whether each individual input has one definite output. It is fine for different inputs to share an output.

Function

A function maps every allowed input to exactly one output. The notation $f:A\to B$ says that inputs come from set $A$ , and their outputs lie in set $B$ . Sending the same input $1$ to both $2$ and $3$ would not define a function.

A function can even send $100$ different inputs to a single output: $f(x)=7$ does this for every one of them. “Exactly one output per input” does not mean “all outputs must differ”.

The notation $f:x\mapsto x^2+1$ reads “ $f$ maps $x$ to $x^2+1$ ”. Equivalently, write $f(x)=x^2+1$ . The expression $f(-2)$ means evaluate the rule with input $-2$ , replacing every $x$ by $-2$ . It is not multiplication by a letter $f$ .

Worked example 1: list a finite mapping

Question. Let $f:x\mapsto x^2+1$ , with allowed inputs $\{-2, 0, 3\}$ . Find the output for each input and list the distinct outputs.

Plan. Substitute each input into the complete rule.

\begin{aligned} f(-2)&=(-2)^2+1=5 &&\text{square the negative input}\\ f(0)&=0^2+1=1 &&\text{substitute zero}\\ f(3)&=3^2+1=10 &&\text{substitute three} \end{aligned}

Why bracket $-2$ ? The whole input is squared, including its negative sign.

Answer. The mappings are $-2\mapsto5$ , $0\mapsto1$ , $3\mapsto10$ . The distinct outputs are $\{1, 5, 10\}$ .

Check. Each of the three allowed inputs has exactly one output, so this is a function.

Change only the input set by adding $2$ . Its output repeats one already reached, but that creates no ambiguity about what each input does.

Worked example 2: two inputs share an output

Question. Let $f:x\mapsto x^2+1$ , now with allowed inputs $\{-2, 0, 2, 3\}$ . Find the outputs and say whether this is a function.

Plan. Keep the old mappings and evaluate the new input.

\begin{aligned} f(-2)&=5 &&\text{unchanged mapping}\\ f(0)&=1 &&\text{unchanged mapping}\\ f(3)&=10 &&\text{unchanged mapping}\\ f(2)&=2^2+1=5 &&\text{evaluate the added input} \end{aligned}

Why is the repeated $5$ allowed? Neither input $-2$ nor input $2$ has two different outputs.

Answer. The outputs are $5, 1, 5, 10$ for inputs $-2, 0, 2, 3$ respectively. This is a function; its distinct outputs are $\{1, 5, 10\}$ .

Check. There are four input arrows, one from each input, although only three distinct outputs are reached.

The diagram includes an unused member $12$ in the output set $B$ . That is allowed: saying outputs lie in $B$ does not promise that every member of $B$ is reached.

[DIAGRAM: asset_slug: m045_finite_mapping; caption: Two inputs share an output, while one member of the output set is unused; description: Function f maps minus 2 and 2 to 5, 0 to 1, and 3 to 10; the output set B also contains 12, which no arrow reaches; file: m045_finite_mapping.png; recommended_method: matplotlib]

Read the notation as an instruction to substitute.

Watch out

The wrong line $f(3)=3f$ treats the notation as multiplication. For $f(x)=x^2+1$ , the correct calculation is $f(3)=3^2+1=10$ .

The Higher formulae sheet does not supply function-notation rules. Use the definition and the rule stated in the question.

Domain and excluded inputs (Higher)

A formula can look perfectly reasonable while failing at a particular input. For $1/(x-2)$ , input $2$ would require division by zero. You must remove it before treating the rule as a real-valued function.

Domain

The domain is the set of allowed inputs. It may be given explicitly, or restricted by the calculation or setting. For example, a number of tickets must be a nonnegative whole number even if an algebraic rule accepts fractions.

Two common arithmetic restrictions are worth checking. A denominator must be nonzero. An expression under a real square root must be nonnegative, including zero because $\sqrt{0}=0$ . These restrictions are not listed on the Higher formulae sheet: learn them.

Worked example 3: exclude a zero denominator

Question. Find the largest real domain of $h(x)=\frac{3}{x+4}$ .

Plan. Find the input that makes the denominator zero and exclude only that input.

\begin{aligned} x+4&=0 &&\text{find the forbidden denominator}\\ x&=-4 &&\text{solve} \end{aligned}

Why exclude $-4$ ? Substituting it would require $3/0$ , which is undefined. Every other real input gives a nonzero denominator.

Answer. All real $x$ such that $x\ne-4$ .

Check. A nearby allowed input $-3$ gives $h(-3)=3$ , whereas $-4$ makes the denominator zero.

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Algebra

Shape, space and measure

Handling data

3.2 - Functions, domain and range

Mappings and function notation (Higher)

Domain and excluded inputs (Higher)

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