Mastering Functions: Domain, Range, Variables, and Function Notation

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

Day, $x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
Number of Visitors, $N$	$1300$	$1500$	$1800$	$1300$	$1900$	$2500$	$1900$

Day,

x

1

2

3

4

5

6

7

Number of Visitors,

N

1300

1500

1800

1300

1900

2500

1900

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

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}

$d$	$15 d$	$k$
$1$	$k = 15 (1)$	$15$
$3$	$k = 15 (3)$	$45$
$5$	$k = 15 (5)$	$75$
$7$	$k = 15 (7)$	$105$

d

15 d

k

1

k = 15 (1)

15

3

k = 15 (3)

45

5

k = 15 (5)

75

7

k = 15 (7)

105

a According to the model, the exterior walls of the pool are

x + 2

meters long,

10

meters wide, and

6

meters tall. Since the wall is one meter thick, the pool inside is

(x + 2) - 2 = x

meters long and

10 - 2 = 8

meters wide. The top of the pool is open, so only

1

meter needs to be subtracted from its height. The water in the pool will be

6 - 1 = 5

meters deep.

The volume of a prism is the product of its dimensions, so the volume of the pool

V

can be written as a function of

x

in this case.

V (x) = x \cdot 8 \cdot 5 \Leftrightarrow V (x) = 40 x

Use the fact that the pool is designed to hold

520

cubic meters of water can be used to find the value of

x .

V (x) = 40 x

Substitute

$V (x) = 520$

520 = 40 x

▼

Solve for

x

DivEqn

$LHS / 40 = RHS / 40$

\frac{5 2 0}{4 0} = x

CalcQuot

Calculate quotient

13 = x

RearrangeEqn

Rearrange equation

x = 13

The amount of concrete needed to construct the pool can also be described as a function of

x .

The amount of cement needed is the difference between the volumes of the outer and inner dimensions of the pool.

Simplify the right-hand side of the function and substitute

x = 13 .

C (x) = (x + 2) \cdot 10 \cdot 6 - (x \cdot 8 \cdot 5)

▼

Simplify right-hand side

Multiply

C (x) = 60 (x + 2) - 40 x

Distr

Distribute $60$

C (x) = 60 x + 120 - 40 x

SubTerm

Subtract term

C (x) = 20 x + 120

Substitute

$x = 13$

C (13) = 20 \cdot 13 + 120

▼

Evaluate right-hand side

Multiply

C (13) = 260 + 120

AddTerms

Add terms

C (13) = 380

The construction of the pool requires

380

cubic meters of concrete.

b The pool has a capacity of

520

cubic meters of water. With a leakage rate of

2

cubic meters per day, the amount of water lost after

d

days can be expressed as

2 d

cubic meters. This means that the remaining amount of water in the pool

R

after

d

days is given by the following function.

R (d) = 520 - 2 d

To determine how many days it will take for the amount of water in the pool to drop to

490

cubic meters, substitute

R (d) = 490

into the equation and solve for

d .

R (d) = 520 - 2 d

Substitute

$R (d) = 490$

490 = 520 - 2 d

▼

Solve for

d

AddEqn

$LHS + 2 d = RHS + 2 d$

490 + 2 d = 520

SubEqn

$LHS - 490 = RHS - 490$

2 d = 30

DivEqn

$LHS / 2 = RHS / 2$

d = 15

This means that after

15

days, there will be

490

cubic meters of water left in the pool.

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Catch-Up and Review

Comparing Graphs of Two Relations

Function

Determining if a Relation Is a Function Using Mapping Diagrams

Visitors to Aquatic Wonders World

Answer

Hint

Solution

Vertical Line Test

Why

Total Number of Fish in the Aquatic Wonders World Aquarium

Hint

Solution

Analyzing Relations

Function Notation

Extra

Domain of a Function

Range of a Function

Exploring Functions in Vending Machines

Answer

Hint

Solution

Identifying Domain and Range

Hint

Solution

Functions as Machines

Independent and Dependent Variables

Evaluating a Function

How Much Fish Can the Dolphins Eat?

Answer

Hint

Solution

Finding the Input of a Function

Construction of New Pool in the Aquarium

Hint

Solution

Relations to Functions

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