Understanding Relations and Functions: Domain, Vertical Line Test, and More

Ages	$18$	$20$	$23$	$20$	$21$	$19$
Heights (m)	$1.70$	$1.96$	$1.85$	$1.75$	$1.91$	$1.87$

Ages

18

20

23

20

21

19

Heights (m)

1.70

1.96

1.85

1.75

1.91

1.87

Ages	$18$	$20$	$23$	$20$	$21$	$19$
Heights (m)	$1.70$	$1.96$	$1.85$	$1.75$	$1.91$	$1.87$

Ages

18

20

23

20

21

19

Heights (m)

1.70

1.96

1.85

1.75

1.91

1.87

Determining Whether a Relation Is a Function
If represented as	Use
a set of coordinates or a table of values	a mapping diagram
a graph in the coordinate plane	the vertical line test

Determining Whether a Relation Is a Function

If represented as

Use

a set of coordinates or a table of values

a mapping diagram

a graph in the coordinate plane

the vertical line test

Function	Analysis	Domain
$f (x) = 3 x$	Multiplication by $3$ is defined for all real numbers.	All real numbers
$g (x) = x$	Square roots are not defined for negative numbers.	All non-negative numbers — that is, $x \geq 0$
$h (x) = \frac{1}{x}$	Division by zero is undefined.	All real numbers except $0$ — that is, $x \neq = 0$

Function

Analysis

Domain

f (x) = 3 x

Multiplication by

3

is defined for all real numbers.

All real numbers

g (x) = x

Square roots are not defined for negative numbers.

All non-negative numbers — that is,

x \geq 0

h (x) = \frac{1}{x}

Division by zero is undefined.

All real numbers except

0

— that is,

x \neq = 0

Function	Domain
$f (x) = 2 x$	All integers
$g (x) = x^{2}$	All real numbers
$h (x) = 4$	All real numbers

Function

Domain

f (x) = 2 x

All integers

g (x) = x^{2}

All real numbers

h (x) = 4

All real numbers

Function	Domain	Analysis	Range
$f (x) = 2 x$	All integers	The function takes any integer input and produces an output that is an even number, as each input is multiplied by $2 .$	All even numbers
$g (x) = x^{2}$	All real numbers	The function takes any real number input and produces an output that is a non-negative number, as each input is squared.	All non-negative numbers — that is, $y \geq 0$
$h (x) = 4$	All real numbers	The function takes any real number input and sends it to $4 .$	Only the number $4$ — that is, the range is ${4}$

Function

Domain

Analysis

Range

f (x) = 2 x

All integers

The function takes any integer input and produces an output that is an even number, as each input is multiplied by

2 .

All even numbers

g (x) = x^{2}

All real numbers

The function takes any real number input and produces an output that is a non-negative number, as each input is squared.

All non-negative numbers — that is,

y \geq 0

h (x) = 4

All real numbers

The function takes any real number input and sends it to

4 .

Only the number

4

— that is, the range is

{4}

Relation I	Relation II	Relation III
$(y - 2) (y + 1) = x$	$y = x (x + 1) (x - 2)$	$x y - 2.25 - 2 - 1.75 0 - 1 2 - 1 - 1 01 1 - 0.75 2 0.5$

Relation I

Relation II

Relation III

(y - 2) (y + 1) = x

y = x (x + 1) (x - 2)

x y - 2.25 - 2 - 1.75 0 - 1 2 - 1 - 1 01 1 - 0.75 2 0.5

a Given a set of coordinate pairs, the domain includes all the first coordinates, and the range includes all the second coordinates.

By listing the numbers from least to greatest, and including each element only once, the domain and range of

R

are as follows.

Domain of R Range of R = {12, 18, 23, 43, 47} = {1.36, 1.75, 1.82, 1.85}

b As done in Part A, to find the domain of

D,

look at the first coordinate of each pair. To find the range, look at the second coordinate of each pair.

By listing the numbers from least to greatest, and including each element only once, the domain and range of

D

are as follows.

Domain of D Range of D = {15, 21, 41} = {1.72, 1.88, 1.91, 1.94}

c Start by drawing a mapping diagram representing the relation given by the sets

R

and

D .

The elements in the domains — the ages — will be grouped in one column, and the elements in the ranges — the heights — will be grouped in a second column.

Then, each age will be connected to the corresponding height using an arrow.

Remember that for mapping diagrams, a relation is a function if there is only one arrow drawn from each input to an output. In this case, the age is the input.

In the first mapping diagram, each age is assigned to only one height. Therefore,

R

represents a function. On the contrary, in the second mapping diagram,

41

is assigned to two different heights. In that case, set

D

cannot represent a function.

Only set $R$ represents a function.

By analyzing the two data sets, some other conclusions can be drawn.

Ramsha's guardians are of different ages and are the same height.
Ramsha has two siblings.
Diego's guardians are both $41$ years old.
Diego has one sibling.

Of course, based on the given information, it does not have to be the exact situation. The conclusions were done simply to show one way of interpreting the data recorded.

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Catch-Up and Review

Answer

Hint

Solution

Table of Values

Mapping Diagram

Set of Ordered Pairs

Points in the Coordinate Plane

Why

Hint

Solution

Hint

Solution