Functions, domains and ranges

A function is a rule that assigns exactly one output to each permitted input. The permitted inputs form the domain. The outputs the function can actually produce form the range.

Functions give precise language for relationships between quantities. They also underpin graphs, transformations, calculus, numerical methods and mathematical modelling.

Before you start

You should be comfortable with:

• substituting values into algebraic expressions
• solving equations and inequalities
• sketching standard graphs
• interval notation and inequality notation

If function notation is completely new, the foundation lesson on function notation provides a shorter introduction.

What makes a relation a function?

Suppose a rule sends an input $x$ to an output $y$. It is a function if every allowed value of $x$ has one and only one output.

For example,

$f(x)=x^2$

is a function. The input $3$ produces the single output $9$. The fact that both $3$ and $-3$ produce $9$ does not cause a problem. Different inputs are allowed to share an output.

By contrast, the relation

$y^2=x$

does not define $y$ as a function of $x$ for $x>0$, because

$y=\sqrt{x}\quad\text{or}\quad y=-\sqrt{x}.$

For example, $x=4$ would have two outputs, $2$ and $-2$.

The crucial test is therefore:

One input cannot be assigned two different outputs.

Function notation

The notation

$f(x)=3x+5$

names the function $f$ and describes its output for an input $x$. The expression $f(x)$ is read as ”$f$ of $x$”. It does not mean $f\times x$.

Example 1: evaluating a function

Let

$f(x)=2x^2-3x+1.$

To find $f(4)$, replace every $x$ with $4$:

$$\begin{aligned} f(4)&=2(4)^2-3(4)+1\\ &=32-12+1\\ &=21. \end{aligned}$$

Brackets protect the substitution, especially when the input is negative.

Example 2: a negative input

Using the same function,

$$\begin{aligned} f(-2)&=2(-2)^2-3(-2)+1\\ &=8+6+1\\ &=15. \end{aligned}$$

The square gives $(-2)^2=4$, not $-4$.

Example 3: an algebraic input

Find $f(a+1)$ when $f(x)=x^2-4x$.

Substitute the entire expression $a+1$:

$$\begin{aligned} f(a+1)&=(a+1)^2-4(a+1)\\ &=a^2+2a+1-4a-4\\ &=a^2-2a-3. \end{aligned}$$

Function notation allows the input to be a number, a variable or a complete expression.

Mappings

A function can be described as a mapping from one set to another:

$f:A\to B.$

This says that $f$ maps elements of the set $A$ into the set $B$. More explicitly,

$x\mapsto f(x)$

means that the input $x$ is mapped to the output $f(x)$.

For example,

$f:\mathbb{R}\to\mathbb{R},\qquad x\mapsto 2x-7$

defines a function from the real numbers to the real numbers.

These two arrows have different jobs:

• $f:A\to B$ names the input and output sets.
• $x\mapsto f(x)$ states what happens to an individual input.

Domain, codomain and range

These terms are related but not interchangeable.

• The domain is the set of allowed inputs.
• The codomain is the declared set into which outputs are mapped.
• The range is the set of outputs actually produced.

For a function $f:A\to B$, the domain is $A$ and the codomain is $B$. The range is a subset of $B$:

$\operatorname{range}(f)\subseteq B.$

Example 4: codomain and range are not always equal

Consider

$f:\mathbb{R}\to\mathbb{R},\qquad f(x)=x^2.$

Its domain is $\mathbb{R}$ and its codomain is also $\mathbb{R}$. However, a square cannot be negative, so its range is

$[0,\infty)$

or equivalently

$\{y\in\mathbb{R}:y\ge0\}.$

The codomain includes negative real numbers, but the function never reaches them.

Finding a domain

Sometimes a domain is stated explicitly. If it is not, use the largest real domain for which the formula makes sense.

At A-level, check especially for:

1. denominators, which cannot equal zero
2. even roots, whose contents must be non-negative
3. logarithms, whose arguments must be positive

Example 5: denominator restriction

For

$f(x)=\frac{5}{x-3},$

the denominator cannot be zero:

$x-3\ne0.$

Therefore the domain is

$x\in\mathbb{R},\qquad x\ne3.$

Example 6: square-root restriction

For

$g(x)=\sqrt{7-2x},$

the expression under the square root must satisfy

$7-2x\ge0.$

Solving gives

$x\le\frac72.$

So the domain is

$\left(-\infty,\frac72\right].$

Remember that dividing an inequality by $-2$ reverses its direction.

Example 7: two restrictions together

Find the domain of

$h(x)=\frac{\sqrt{x+1}}{x-4}.$

The square root requires

$x+1\ge0\quad\Longrightarrow\quad x\ge-1.$

The denominator requires

$x\ne4.$

Both conditions must hold, so the domain is

$[-1,4)\cup(4,\infty).$

Finding a range

There is no single method for every function. Useful approaches include:

• using the shape and turning points of a graph
• writing the formula in a form that exposes its minimum or maximum
• setting $y=f(x)$ and asking which values of $y$ give a real solution for $x$
• using known ranges of standard functions

Example 8: range of a quadratic

Find the range of

$f(x)=x^2-6x+11,$

where $x\in\mathbb{R}$.

Complete the square:

$f(x)=(x-3)^2+2.$

Since

$(x-3)^2\ge0,$

we have

$f(x)\ge2.$

The value $2$ occurs when $x=3$, so the range is

$[2,\infty).$

Example 9: range on a restricted domain

Let

$g(x)=x^2-6x+11,qquad 0\le x\le4.$

The completed-square form is still

$g(x)=(x-3)^2+2.$

The minimum occurs at $x=3$ and is $2$. To find the maximum on the closed interval, check the endpoints:

$g(0)=11,qquad g(4)=3.$

Therefore the range is

$[2,11].$

Checking only the turning point would miss the maximum.

Example 10: range by rearranging

Find the range of

$f(x)=\frac{2x+1}{x-3},qquad x\ne3.$

Set

$y=\frac{2x+1}{x-3}.$

Rearrange for $x$:

$y(x-3)=2x+1,$

$x(y-2)=3y+1,$

$x=\frac{3y+1}{y-2}.$

This expression exists for every real $y$ except $y=2$. Hence the range is

$y\in\mathbb{R},\qquad y\ne2.$

One-to-one and many-to-one functions

A function is one-to-one if different inputs always produce different outputs:

$f(a)=f(b)\quad\Longrightarrow\quad a=b.$

For example, $f(x)=3x+1$ on $\mathbb{R}$ is one-to-one.

A function is many-to-one if at least two different inputs share an output. For $f(x)=x^2$ on $\mathbb{R}$,

$f(2)=f(-2)=4,$

so it is many-to-one.

Both are valid functions. The distinction concerns whether outputs are repeated, not whether each input has one output.

The horizontal line test

A graph represents a one-to-one function if every horizontal line meets it at most once. If a horizontal line meets it more than once, the function is many-to-one.

This matters because only a one-to-one function can be reversed without ambiguity. Domain restrictions and inverse functions are developed in composite and inverse functions.

Example 11: proving a linear function is one-to-one

Let $f(x)=5x-4$. Suppose $f(a)=f(b)$. Then

$5a-4=5b-4,$

so

$5a=5b$

and therefore

$a=b.$

Thus $f$ is one-to-one on $\mathbb{R}$.

Solving equations written in function notation

Example 12: find an input from an output

Let

$f(x)=x^2+2x-3.$

Solve $f(x)=5$.

Substitute the formula for $f(x)$:

$x^2+2x-3=5,$

$x^2+2x-8=0,$

$(x+4)(x-2)=0.$

Hence

$x=-4\quad\text{or}\quad x=2.$

Two inputs produce the output $5$, which also shows that this quadratic is many-to-one on $\mathbb{R}$.

Common misconceptions

• Thinking $f(x)$ means $f\times x$. It names the output of function $f$ at input $x$.
• Objecting when two inputs share an output. That is allowed. A function fails only when one input has more than one output.
• Using domain, codomain and range as synonyms. The range contains outputs actually achieved; the codomain is the declared target set.
• Ignoring a stated domain. A restriction can change the range and whether a function is one-to-one.
• Cancelling away restrictions. Even if an algebraic expression simplifies, excluded values from the original rule remain excluded.
• Finding a quadratic range from the constant term. Use the turning point and any domain endpoints.
• Forgetting brackets in substitutions. Write $f(-2)$ by replacing every $x$ with $(-2)$ before simplifying.

Quick checks

1. Evaluate a function

Given $f(x)=3x^2-2x$, find $f(-1)$ and $f(2a)$.

Answer

$f(-1)=3(-1)^2-2(-1)=5.$

$f(2a)=3(2a)^2-2(2a)=12a^2-4a.$

2. Find a domain

Find the domain of

$g(x)=\sqrt{3x-6}.$

Answer

Require $3x-6\ge0$, so $x\ge2$. The domain is $[2,\infty)$.

3. Find a range

Find the range of

$h(x)=4-(x+1)^2,$

where $x\in\mathbb{R}$.

Answer

Since $(x+1)^2\ge0$,

$4-(x+1)^2\le4.$

The maximum $4$ occurs at $x=-1$, so the range is $(-\infty,4]$.

4. Classify a function

Is $f(x)=x^3$ one-to-one or many-to-one on $\mathbb{R}$?

Answer

It is one-to-one. The function is strictly increasing, so different real inputs cannot produce the same output. Every horizontal line meets its graph exactly once.

What to learn next

Next, study composite and inverse functions to combine and reverse function rules. Then use function graphs and graph transformations to connect algebraic rules with their geometry.