Relations, Functions, and the Vertical Line Test, precalculus lesson

A class list pairs every student with a date of birth. Two students may share a birthday, but no student has two. That one-sided rule is exactly what separates a function from any other way of pairing things, and on a graph you can see it with a single vertical line.

Key terms

Ordered pair

Definition An ordered pair $(a, b)$ consists of a first coordinate $a$ and a second coordinate $b$ , in that order. Two ordered pairs are equal exactly when their first coordinates are equal and their second coordinates are equal,

\begin{aligned} &(a, b) = (c, d) \\[0.4em] &\iff a = c \text{ and } b = d. \end{aligned}

The same notation $(a, b)$ is also used for an open interval. The context tells which one is meant.

In plain words In a pair, order matters. The pair $(2, 5)$ and the pair $(5, 2)$ are different, while the sets $\{2, 5\}$ and $\{5, 2\}$ are the same.

Example Suppose $(x - 1, 4) = (3, y + 2)$ . Comparing first coordinates gives $x - 1 = 3$ , and comparing second coordinates gives $4 = y + 2$ . So $x = 4$ and $y = 2$ .

Coordinate plane

Definition The coordinate plane is a plane with two perpendicular real number lines, the horizontal $x$ -axis and the vertical $y$ -axis, which cross at their common zero point, the origin. Positive numbers lie to the right on the $x$ -axis and upward on the $y$ -axis. Each point $P$ is matched with exactly one ordered pair $(x, y)$ of real numbers, and each ordered pair of real numbers with exactly one point. The number $x$ is where the vertical line through $P$ crosses the $x$ -axis, and $y$ is where the horizontal line through $P$ crosses the $y$ -axis. In this way the plane is identified with the set

\mathbb{R}^2 = \{(x, y) \mid x \in \mathbb{R} \text{ and } y \in \mathbb{R}\}.

The axes split the rest of the plane into four quadrants, numbered I to IV counterclockwise starting at the upper right. Points on an axis belong to no quadrant.

Quadrant	Sign of $x$	Sign of $y$
I	positive	positive
II	negative	positive
III	negative	negative
IV	positive	negative

In plain words The first coordinate says how far to go right or left from the origin, and the second says how far to go up or down.

Example To plot $(3, -2)$ , start at the origin, move 3 units right and then 2 units down. The point lies in quadrant IV. The point $(-2, 3)$ lies in quadrant II, so swapping the coordinates moved the point somewhere else entirely. The point $(0, -5)$ lies on the $y$ -axis and in no quadrant.

Relation

Definition For sets $A$ and $B$ , write $A \times B$ for the set of all ordered pairs with first coordinate in $A$ and second coordinate in $B$ ,

\begin{aligned} &A \times B \\[0.4em] &\quad = \{(a, b) \mid a \in A \text{ and } b \in B\}. \end{aligned}

A relation from $A$ to $B$ is any subset $R$ of $A \times B$ . When $(a, b) \in R$ , we say that $a$ is related to $b$ . A relation between real numbers is a set of ordered pairs of real numbers, and its graph is the set of all points in the coordinate plane with those coordinates.

In plain words A relation is a collection of pairs that says which first items go with which second items. Any collection of pairs counts, with no further conditions.

Example Let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$ . The set $R = \{(1, 4), (1, 6), (3, 5)\}$ is a relation from $A$ to $B$ . In it, $1$ is related to two elements and $2$ to none. An equation can define a relation too. The pairs $(x, y)$ of real numbers with $x^2 + y^2 = 25$ form a relation whose graph is the circle of radius $5$ centered at the origin. It contains $(3, 4)$ , $(3, -4)$ and $(-5, 0)$ .

Function

Definition A function $f$ from a set $A$ to a set $B$ , written $f \colon A \to B$ , is a relation from $A$ to $B$ in which every element of $A$ is the first coordinate of exactly one ordered pair in $f$ . If $(a, b) \in f$ , then $b$ is called the value of $f$ at $a$ .

“Exactly one” contains two requirements. Every element of $A$ has at least one partner, and no element of $A$ has two different partners. When no sets $A$ and $B$ are named, a set of ordered pairs is simply called a function if no first coordinate appears in two pairs with different second coordinates. It is then a function from the set of its first coordinates.

In plain words A function gives each allowed input one and only one output.

Example Again let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$ . The set

f = \{(1, 4), (2, 6), (3, 4)\}

is a function from $A$ to $B$ , since $1$ , $2$ and $3$ each appear exactly once as a first coordinate. The relation $R$ from the previous section is not a function from $A$ to $B$ , because $1$ has two partners and $2$ has none.

Domain

Definition The domain of a relation is the set of all first coordinates of its pairs. For a function $f \colon A \to B$ the domain is the whole set $A$ , and every element of the domain has exactly one value. The set of all values is the range of $f$ , which has a lesson of its own. When a function of real numbers is given by a formula and no domain is stated, the domain is the set of all real numbers for which the formula gives a real number, called the natural domain.

In plain words The domain is the set of inputs the function accepts, and the function must have an answer for every one of them.

Example The function $\{(-1, 1), (0, 0), (1, 1), (2, 4)\}$ has domain $\{-1, 0, 1, 2\}$ and range $\{0, 1, 4\}$ . The pairs $(x, \sqrt{x})$ with $x \ge 0$ form a function with domain $[0, \infty)$ . No pair starts with $-4$ , because $\sqrt{-4}$ is not a real number.

Inputs, outputs and variables

Definition For a function $f$ , an input is an element $x$ of the domain, and the output for $x$ is the unique $y$ with $(x, y) \in f$ . Suppose a function is described by an equation solved for $y$ , such as $y = x^2 + 1$ , which gives exactly one $y$ for each allowed $x$ . Then $x$ is called the independent variable, since it can take any value in the domain, and $y$ is called the dependent variable, since its value is determined by $x$ . The graph of the function is the set of all points $(x, y)$ where $y$ is the output for $x$ , with the independent variable on the horizontal axis.

In plain words You choose the input, and the function decides the output. The independent variable is the one you are free to choose.

Example A square with side length $s$ centimeters has area

A = s^2 \text{ square centimeters.}

Here $s$ is the independent variable and $A$ is the dependent one. The input $s = 3$ gives the output $A = 9$ , so the pair $(3, 9)$ is on the graph. Only positive side lengths make sense for a real square, so in this situation the domain is $(0, \infty)$ .

Mapping diagram

Definition A mapping diagram of a relation from a finite set $A$ to a finite set $B$ lists the elements of $A$ in one column and the elements of $B$ in a second column, and draws an arrow from $a$ to $b$ for each pair $(a, b)$ in the relation. The relation is a function from $A$ to $B$ exactly when every element of $A$ has exactly one arrow leaving it. Elements of $B$ may receive any number of arrows, including none.

In plain words Count the arrows leaving each input. One arrow from every input means a function, whatever happens on the output side.

Example Take $A = \{1, 2, 3\}$ , $B = \{7, 8, 9\}$ and the relation $\{(1, 7), (2, 7), (3, 9)\}$ . Its arrows are

1 \to 7, \qquad 2 \to 7, \qquad 3 \to 9.

Each input has exactly one arrow, so this is a function. The output $7$ receives two arrows and $8$ receives none, and both are allowed. Adding the pair $(3, 8)$ would give $3$ a second arrow, and the relation would no longer be a function.

Vertical line test

Definition A set $G$ of points in the coordinate plane is the graph of a function of $x$ exactly when every vertical line meets $G$ in at most one point. In that case the domain of the function is the set of all numbers $c$ for which the vertical line $x = c$ meets $G$ . The reason is that the points of $G$ on the line $x = c$ are exactly the pairs in $G$ with first coordinate $c$ .

In plain words Slide a vertical line across the graph. If it ever hits the graph twice, some input has two outputs, and the graph does not belong to a function.

Example On the circle $x^2 + y^2 = 25$ , the vertical line $x = 3$ meets the points $(3, 4)$ and $(3, -4)$ , since $9 + 16 = 25$ . So the circle is not the graph of a function of $x$ . On the parabola $y = x^2$ , the line $x = c$ meets only the point $(c, c^2)$ , so the parabola passes. The upper half of the circle, given by

y = \sqrt{25 - x^2},

also passes, and its domain is $[-5, 5]$ . A sketch only shows part of a graph, so for a graph given by an equation, check the test with algebra as done here.

Many-to-one versus one-to-many

Definition A relation is many-to-one when at least two different first coordinates share the same second coordinate. It is one-to-many when at least one first coordinate is paired with two or more different second coordinates. A function may be many-to-one, but a function is never one-to-many, since that would give an input two outputs. A function in which different inputs always have different outputs is called one-to-one.

In plain words Many inputs sharing one output is fine for a function. One input with many outputs is not.

Example The function $y = x^2$ is many-to-one, because the inputs $-2$ and $2$ both give the output $4$ . The relation $y^2 = x$ is one-to-many, because the input $x = 4$ is paired with both $y = 2$ and $y = -2$ . So $y^2 = x$ does not define $y$ as a function of $x$ . Read the other way, with $y$ as the input, the same equation $x = y^2$ does define $x$ as a function of $y$ .

Check yourself

Is $\{(2, 3), (4, 3), (6, 5)\}$ a function? Give its domain and say whether it is many-to-one, one-to-many or neither.

Show the answer

Yes. Each of $2$ , $4$ and $6$ appears exactly once as a first coordinate, so it is a function with domain $\{2, 4, 6\}$ . It is many-to-one, since the inputs $2$ and $4$ share the output $3$ .

Does the equation $y^2 = x + 1$ define $y$ as a function of $x$ ?

Show the answer

No. For $x = 3$ the equation becomes $y^2 = 4$ , so both $y = 2$ and $y = -2$ work. The vertical line $x = 3$ meets the graph at $(3, 2)$ and $(3, -2)$ .

For which real numbers $k$ is $\{(1, 2), (3, 5), (k, 7)\}$ a function?

Show the answer

For every real $k$ except $1$ and $3$ . If $k = 1$ , the input $1$ has the two outputs $2$ and $7$ . If $k = 3$ , the input $3$ has the two outputs $5$ and $7$ . Any other $k$ is a new input with a single output.

Relations, Functions, and the Vertical Line Test

Theory, part 3

Practice, part 4

Key terms

Ordered pair

Coordinate plane

Relation

Function

Domain

Inputs, outputs and variables

Mapping diagram

Vertical line test

Many-to-one versus one-to-many

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus