Domains, Ranges, and One-One Functions

Overview

This branch develops the language needed before inverse and composite functions make sense.

Use it with the hub Functions.

The core idea is:

• a relation may connect inputs and outputs in many ways;
• a function is a relation with exactly one output for every allowed input;
• domain and range describe the input and output behaviour of that mapping;
• one-one behaviour asks whether outputs identify inputs uniquely.

Set Notation

Domains and ranges are sets.

Useful notation:

• $x\in (a,b)$ means $a<x<b$.
• $x\in [a,b)$ means $a\le x<b$.
• $\{x\in \mathbb{R}:x\ge 0\}$ means all real $x$ such that $x\ge 0$.
• $\mathbb{R}\setminus \{0\}$ means all real numbers except $0$.
• $A\cup B$ means union.

Common number sets:

• $\mathbb{R}$: real numbers;
• ${\mathbb{R}}^{+}$: positive real numbers;
• ${\mathbb{R}}^{-}$: negative real numbers;
• ${\mathbb{R}}^{+}\cup \{0\}$: non-negative real numbers.

Relations and Functions

A relation is an association between inputs and outputs.

It can be represented:

• numerically, using a table of values;
• graphically, using a curve;
• diagrammatically, using mapping arrows;
• algebraically, using an equation or rule.

A relation is a function if every allowed input has exactly one output.

How to read this figure. Follow the outgoing arrows from each domain element. In panel A every input has exactly one output, even though $b$ and $c$ share $y_2$; this is a valid many-to-one function. In panel B, $b$ points to both $y_2$ and $y_3$, so one input has two outputs and the relation is not a function.

Important distinctions:

• Two different inputs may share the same output and the relation can still be a function.
• One input with two different outputs means the relation is not a function.
• An input in the domain with no output also means the relation is not a function.

Rule, Domain, Codomain, and Range

A function can be written as

$$f:X\to Y.$$

Here:

• $X$ is the domain $D_f$, the set of allowed inputs;
• $Y$ is the codomain, the declared target set;
• the rule tells how each $x\in D_f$ is mapped;
• the range $R_f$ is the set of actual outputs.

The range is

$$R_f=\{f(x):x\in D_f\}.$$

How to read this figure. The outer right-hand box is the declared codomain. The dashed region contains the values actually reached by arrows and is therefore the range. Unreached values such as $y_1$ and $y_4$ remain in the codomain, showing why codomain and range need not be equal.

For example,

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

has

$$D_f=\mathbb{R},R_f=[0,\infty ).$$

The codomain is $\mathbb{R}$, but the range is only $[0,\infty )$.

Equality of Functions

Two functions are equal only if they have the same:

• domain;
• codomain, when the codomain is part of the stated function;
• output for every input in the domain.

So a function is not determined by its formula alone.

For example,

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

and

$$g:[0,\infty )\to \mathbb{R},g(x)=x^2$$

are different functions because their domains are different.

This is not a technicality. It changes whether the function is one-one and whether an inverse function exists.

Maximal Domain

If no domain is given, the convention is to use the maximal real domain: all real values for which the rule makes sense and gives a real output.

Typical restrictions:

• denominator cannot be zero;
• an even root requires the expression inside to be non-negative;
• a logarithm requires the argument to be positive.

Examples:

$$f(x)=\frac{1}{x}$$

has maximal domain

$$\mathbb{R}\setminus \{0\}.$$

Also,

$$g(x)=\ln (x+1)$$

has maximal domain

$$(-1,\infty ).$$

Finding Range

The range is the set of possible outputs on the stated domain.

A safe strategy:

1. identify the domain carefully;
2. sketch or reason from the graph;
3. check turning points and endpoint behaviour;
4. consider asymptotic behaviour if relevant;
5. write the output set in interval notation.

Endpoint checking alone is incomplete. A graph can turn inside the interval.

Example:

$$h(x)=x^2,x\in [-1,2).$$

The minimum value is $0$ at $x=0$, which lies inside the domain. The value $4$ is approached as $x\to 2^{-}$ but not attained.

Hence

$$R_h=[0,4).$$

How to read this figure. The domain restriction $-1\le x<2$ selects only part of the parent parabola. The interior turning point gives the attained minimum $y=0$. The open endpoint at $x=2$ makes $y=4$ only a limiting value, not an attained output. Hence $R_h=[0,4)$.

A reliable range workflow

For a continuous curve on one or more domain intervals:

1. mark included and excluded domain endpoints;
2. locate stationary or turning points inside the domain;
3. inspect asymptotes and end behaviour;
4. decide whether each boundary output is attained or merely approached;
5. combine the output intervals.

For example, an open endpoint in the domain does not automatically produce an open endpoint in the range: another allowed input may still produce the same output. Attainment must be checked from the whole domain.

Piecewise Functions

A piecewise function is one function whose rule changes on different parts of its domain.

Example:

$$g(x)=\{\begin{array}{ll}-x+1,& x<0,\\ x,& x\ge 0.\end{array}$$

The domain is

$$D_g=\mathbb{R}.$$

For range, treat each domain piece separately:

• if $x<0$, then $g(x)=-x+1>1$, so this piece gives $(1,\infty )$;
• if $x\ge 0$, then $g(x)=x\ge 0$, so this piece gives $[0,\infty )$.

Combining the pieces,

$$R_g=[0,\infty ).$$

Vertical Line Test

The vertical line test checks whether a graph represents a function.

• If every vertical line cuts the graph at most once, the graph represents a function.
• If one vertical line cuts the graph more than once, the graph does not represent a function.

Reason: a vertical line fixes one input $x$. More than one intersection means that one input has more than one output.

To show that a relation is not a function, it is enough to provide one counterexample, such as a vertical line $x=a$ that intersects the graph twice.

If a domain has been declared separately, also confirm that every $x$ in that domain has an image. “At most once” rules out multiple outputs; the definition additionally requires one output for every allowed input.

One-One Functions

A function is one-one, or injective, if no two distinct inputs have the same output.

Equivalent tests:

• algebraically, $f(x_1)=f(x_2)$ implies $x_1=x_2$;
• graphically, every horizontal line cuts the graph at most once;
• conceptually, each output identifies at most one input.

How to read this figure. The left panel uses a vertical line because fixing $x$ tests the number of outputs belonging to one input. The right panel uses a horizontal line because fixing $y$ tests how many inputs produce one output. The parabola passes the function test but fails the one-one test on the displayed domain.

Examples:

• $f(x)=x^2$ on $\mathbb{R}$ is not one-one because $f(-1)=f(1)$.
• $f(x)=x^2$ on $[0,\infty )$ is one-one.
• $h(x)=\frac{x-1}{x}$ on $x>0$ is one-one because its graph is crossed at most once by each horizontal line in its range.

Restricting the Domain

If a function is not one-one, it may be restricted to a smaller domain to make it one-one.

The restricted function keeps the same rule but uses a smaller input set.

Example:

$$f(x)=x^2$$

on $\mathbb{R}$ is not one-one. To make it one-one while preserving the range $[0,\infty )$, use either

$$x\ge 0$$

or

$$x\le 0.$$

For

$$g(x)=x^2-4x+1=(x-2{)}^2-3,$$

a largest one-sided domain that makes the function one-one is

$$[2,\infty )$$

or

$$(-\infty ,2].$$

The choice depends on the question.

“Maximal” means that the restricted domain cannot be enlarged while keeping the required one-one behaviour and, when requested, the same range. It does not always mean that the answer is unique. For $x^2$ on $\mathbb{R}$, both $[0,\infty )$ and $(-\infty ,0]$ are maximal one-sided restrictions with range $[0,\infty )$.

Algebraic Test for One-One Behaviour

When a graph is inconvenient, start with two allowed inputs $x_1,x_2$ and assume

$$f(x_1)=f(x_2).$$

If this assumption necessarily leads to $x_1=x_2$, then $f$ is one-one on the stated domain. If it permits distinct inputs, the function is not one-one.

For example, if $f(x)=3x-5$, then

$$3x_1-5=3x_2-5⟹x_1=x_2,$$

so $f$ is one-one on $\mathbb{R}$. For $g(x)=x^2$ on $\mathbb{R}$, $g(1)=g(-1)$ gives an immediate counterexample.

Common Pitfalls

• Thinking that a function must map different inputs to different outputs. That is one-one behaviour, not function behaviour.
• Forgetting that a function is defined by both rule and domain.
• Confusing codomain with range.
• Finding range from endpoint values only.
• Ignoring open endpoints when stating ranges.
• Treating a piecewise function as several unrelated functions.
• Using the horizontal line test before first checking that the relation is already a function.

Related

• Functions
• Inverse Functions
• Composite Functions
• Graphing Techniques