5.4 Domain and Range

Leanne Thompson

5.4 Domain and Range

Another way to analyze a relation is to find the domain and range of the graph. To understand domain and range, first recall that in the ordered pair, (x, y), the x-coordinate is called the input, and the y-coordinate is called the output. The domain of a relation will be the set of all the inputs (x-values) of that relation. The range of a relation will be the set of all the outputs (y-values) of that relation.

Example 1

Identify the domain and range of the relation containing the following set of points:

(2, 1), (2, 4), (2, 6), (4, 5), (5, 4), (6, 2), (6, 4)

Table 5.4.1
Steps	Solution
Find the domain by identifying all the inputs (x-values) of the ordered pairs. If an input appears more than once in the relation, you only need to include it once in the domain.	domain = {2, 4, 5, 6}
Find the range by identifying all the outputs (y-values) of the ordered pairs. If an output appears more than once in the relation, you only need to include it once in the range.	range = {1, 2, 4, 5, 6}

Practice 1

Identify the domain and range of the relations containing the following sets of points.

a) (3, 6), (1, 2), (5, 3)

b) (2, 4), (0, 2), (2, 3)

c) (1, 1), (1, –1), (-3, 5), (2, –5)

Practice 2

Identify the domain and range of the relations represented by the tables of values below.

a)

x	y
–2	–1
0	–5
1	3
3	2

b)

x	y
–2	–4
0	–2
2	0
4	2

Practice 3

Find the domain and range of the relations represented by the graphs below.

a)

Graph with points (2,2),(2,3),(5,-1),(-3,4),(-2,-5),(4,0)

b)

Graph with the points (1,2),(2,3),(3,4),(4,5),(-1,0),(0,1)

c)

A graph of the equation y = 1 third x−1.

d)

Linear graph y=-2x-1 that stops at point (1,-3)

e)

Circle graph x spans -2 to 2 and y spans -2 to 2.

f)

Parabola shaped (u-shape) graph opening up with x-ints at (-1,0) and (1,0) and y-int at (0,1)

Practice 4

Mitchell wants to do some landscaping and plant a garden for his mom, and he figures it will cost around $850 for the whole project. Mitchell has already saved $250, and he plans on saving $50 per week until he reaches his goal. The equation $y=50x+250$ represents the relationship between the number of weeks that have passed, x, and how much Mitchell has saved, y, in dollars.
a) Create a table of values with six relevant points.
b) Label each axis and give the graph a title.
c) Plot the points, then draw a line to connect the points.

x

y

–

Quadrant 1 of a blank coordinate plane. The x-axis has a minimum of 0 and maximum of 16 and a scale of 0.5. The y-axis has a minimum of 0 and maximum of 1200 and a scale of 50.

d) How many weeks will it take for Mitchell to save enough money for the project?

e) What are the domain and range of the graph?

Homework

Identify the domain and range of the relations containing the following sets of points.

a) (2, 3), (3, 4), (4, –5), (2, –2)

b) (1, 0), (2, 5), (3, –3), (4, 5)

c) (–2, 3), (0, 4), (–2, –2), (–2, –1)

d) (5, 6), (7, –3), (7, 6), (7, 5)

Identify the domain and range of the relations using the tables of values below.

a)

x	y
0	4
2	2
4	0
6	–2

b)

x	y
–2	4
–1	1
0	0
1	1

c)

x

y

–5

–10

–3

–6

–1

–2

1

2

d)

x

y

–3

–27

–2

–8

–1

0

Find the domain and range of the relations represented by the graphs below.

a)
b)
c)

d)
e)
f)

g)
h)
i)
Jin brought his baby to a spray park. There is a water feature at the spray park that sprays out water. The graph shows the path of the water before it hits the ground. The d-axis represents the horizontal distance of the water from the water feature in feet, and the h-axis represents the height of the water from the ground in feet. Answer the following questions about the graph.

a) At what height from the ground is the water being sprayed from the water feature?

b) Given that the equation of the path of the water can be represented with the equation $y=-1.3{(x-2)}^2+5.2$ , what is the maximum height that the water will reach in the air?

c) What are the d-intercepts? What do they represent?

d) What are the domain and range of the graph?

Answers

1.

a) Domain = {2, 3, 4} Range = {3, 4, -5, -2}	b) Domain = {1, 2, 3, 4} Range = {0, 5, -3}
c) Domain = {-2, 0} Range = {3, 4, -2, -1}	d) Domain = {5, 7} Range = {6, -3, 5}

2.

a) Domain = {0, 2, 4, 6} Range = {4, 2, 0, -2}	b) Domain = {-2, -1, 0, 1} Range = {4, 1, 0}
c) Domain = {-5, -3, -1, 1} Range = {-10, -6, -2, 2}	d) Domain = {-3, -2, -1, 0} Range = {-27, -8, -1, 0}

3.

a) Domain = {1, 2, 3, 4} Range = {3, 4, 5}	b) Domain = {-3, -2, -1, 0, 1} Range = {5, 4, 3, 2, 1}
c) Domain = { $x\in\mathbb{R}$ } Range = { $y\in\mathbb{R}$ }	d) Domain = { $x\leq 1.5, x\in\mathbb{R}$ } Range = { $y\leq 2,y\in\mathbb{R}$ }
e) Domain = { $x\geq -1,x\in\mathbb{R}$ Range = { $y\in\mathbb{R}$ }	f) Domain = { $x\in\mathbb{R}$ } Range = { $y=-2$ }
g) Domain = { $x\in\mathbb{R}$ } Range = { $y\in\mathbb{R}$ }	h) Domain = { $x\geq 0,x\in\mathbb{R}$ } Range = { $y\geq 0,y\in\mathbb{R}$ }
i) Domain = { $x\in\mathbb{R}$ } Range = { $y\leq 1,y\in\mathbb{R}$ }

4.

a) 0 ft	b) 5.2 ft
c) (0,0) – This is the point the water leaves the water feature. (4,0) – This is the point the water hits the ground.	d) Domain = { $0\leq x\leq 4,x\in\mathbb{R}$ } Range = { $0\leq y\leq 5.2,y\in\mathbb{R}$ }

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0C

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