Function - Domain & Range

Domain and Range

By Kaitlyn Savage

D​omain Definition

​

the domain is the set of inputs accepted by the function

​

our x-values make up the domain

​

We can scan the x-axis from left to right to find our domain.

Range Definition

​The range is our output values accepted by the function

​

therefore, y-values make up the range ​

Scan the y-axis from the bottom to the top to determine range.

​If our function is 2x and our domain is {1,2,3,4}, we substitute each of those x-values into the function to get output values. These output values make up our range of {2,4,6,8}

Notation is IMPORTANT​

​For discrete data (just points plotted on a graph that are not connected), we list the x and y values in braces.

For example, D: {2,4,6}

​

For a line segment with definite endpoints and an interval of values, we list the minimum x/y value and the maximum x/y value in ​brackets to show that it is inclusive.

For example, R: [-4,7]​

​

We use parentheses when there is a hole in the graph and that point is not included.​ We mostly see this used with infinity when a graph will continue on forever since this is just an idea and infinity is not a definite endpoint.

For example, D: (-​∞, ∞)

Multiple Choice

What notation should we use if we just have POINTS? Suppose we are given a few values of x and a few values of y. Think: discrete data

• Parentheses ( )

• Brackets [ ]

• Braces { }

Multiple Choice

What notation should we use to represent INCLUSIVITY? There is an interval of numbers and we note the minimum and maximum.

• Parentheses ( )

• Brackets [ ]

• Braces { }

Multiple Choice

What notation is used to show that there is NOT a definite endpoint there? It just goes UP to that value. This is most commonly used with infinity signs.

• Parentheses ( )

• Brackets [ ]

• Braces { }

Let's Try Some Problems!​

Suppose we are given an input/output table

​Remember: the domain consists of our x-values and the range represents our y-values.

​

Therefore, the domain in this example is {5,6,7} and the range is {3,4,5}

​

This means there is a point at (5,3) (6,4) and (7,5)​

​

​x	​y
​5	​3
​6	​4
​7	​5

What if we are given a graph?​

​When analyzing graphs, scan the x axis from left to right to find domain. Scan the y axis from bottom to top to find range.

In this example, the minimum x value is -1 and the maximum is 1. Therefore, the domain is [-1,1]. The minimum y value is -1 and the maximum is 1. Therefore the range is exactly the same: [-1,1]​

Subject | Subject

Sometimes our graphs will have arrows, meaning it will continue forever. If the x and y values will continue to get smaller as the graph goes left and down, and they will continue to get larger as the graph goes right and up, the domain and range is infinite.

Practice writing infinity signs because we will see this a lot!​

No set endpoints?!​

Open Ended

Find the domain of this graph.

Since it is a line segment with definite ends and there is an INTERVAL of points, we use brackets. Find the minimum and maximum x values and separate them with a comma inside brackets.

Fill in the Blanks

Poll

How are we feeling? Be honest! It helps me as your teacher :)

• I understand, I think I am ready to build on this knowledge!

• I am doing okay but I would like more practice!

• Not great but I could get there if we spend another day or 2 learning this!

• I think this is very hard. I need to see this a lot more to understand it!

Open Ended

Tell me what you think was the hardest part of this lesson

Multiple Choice

What is the domain?

• x = 2
• −∞ < x < ∞
• y = 2
• none of these

Multiple Choice

What is the RANGE?

• {-3, 3]
• all real numbers
• [-4, 5]
• [-4, ∞)

Multiple Choice

What is the DOMAIN?

• [0, ∞)
• (-∞, 0)
• All real numbers
• {0, 1, 2, 3, 4,...}

Multiple Choice

What is the range of the function?

• [-1, +∞)
• (-1,1)
• [1.5, +∞)
• (-2, 2)

Multiple Choice

What is the range? 

• [-∞, +∞]
• [-2, 2]
• (-∞, ∞)
• [0, 2]

Multiple Choice

Does the graph represent a function?
If you have an x value that has 2 "y" values it is NOT a function! 

• Yes, because the vertical line test shows there are no repeating input values
• No, because the vertical line test shows there are repeating input values

Multiple Choice

Which graph does NOT pass the vertical line test?

• Graph 1
• Graph 2
• Graph 3
• Graph 4

Multiple Choice

Is the relation shown a function?

• No, because the x-value 11 is paired with 5 and 8.

• Yes, because each x-value has only one y-value paired with it.

Multiple Choice

Find the Range.

• (-∞, +∞)

• (0, 11)

• [0, ∞)

• (0, ∞)

Multiple Choice

What is the Domain of this linear function?

• [-6, 6]

• [0, 7]

• [2, 7]

• No Solution

Multiple Choice

the set of all possible x-values

• Domain

• Range

• Interval Notation

• Inequality Notation

Multiple Choice

the set of all possible y-values

• Domain

• Range

• Interval Notation

• Inequality Notation

Multiple Choice

What is the domain in Interval notation?

• [ -2 , 2 ]

• [ 4 , -4 ]

• (-2 , 2 )

• (4 , -4 )

Multiple Choice

What is the domain in inequality notation?

• [ -2 , 2 ]

• [ 4 , -4 ]

• -2 ≤ x ≤ 2

• 4 ≥ y ≥ -4

Multiple Choice

What is the range in interval notation?

• [ -2 , 2 ]

• [ 4 , -4 ]

• -2 ≤ x ≤ 2

• 4 ≥ y ≥ -4

Multiple Choice

What is the domain in interval notation?

• ( -5 , 5 )

• [-2 , -4 )

• -5 < x < 5

• -2 ≥ y > -4

Multiple Choice

What is the domain in interval notation?

• [ -2 , 2 )

• ( 5 , 1 ]

• -2 ≤ x < 2

• 5 > y ≥ 1

Multiple Choice

What is the range in interval notation?

• [ -2 , 2 )

• ( 5 , 1 ]

• -2 ≤ x < 2

• 5 > y ≥ 1

Multiple Choice

What is the domain in interval notation?

• ( -4 , 4 )

• [ 1 , -3 )

• [-4, 4]

• (-3, 1]

Multiple Choice

What is the domain in inequality notation?

• ( -4 , 4 )

• [ 1 , -3 )

• -4 < x < 4

• 1 ≥ y > -3

Multiple Choice

Is the following a function?

• YES

• NO

Multiple Choice

DOES THIS GRAPH REPRESENT A FUNCTION?

• NO

• YES

Multiple Choice

DOES THE GRAPH REPRESENT A FUNCTION?

• YES

• NO

Multiple Select

• A

• B

• C

• D

Multiple Choice

Is the relation a function? Why.

• Yes, because the x-value 11 has two y-values pair with it.
• Yes, because each x-value has only one y-value paired with it.
• No, because the x-value 11 has two y-values pair with it.
• No, because each x-value has only one y-value paired with it.

Multiple Choice

You can identify a function from a graph by using the.....

• Horizontal Line Test
• Eye Test
• Vertical Line Test
• Diagonal Line Test

Multiple Choice

Which set of values is a function?

• (9,5) (10,5) (9,-5) (10,-5)
• (3,4) (4,-3) (7,4) (3, 8)
• (6,-5) (7, -3) (8, -1) (9, 1)
• (2, -2) (5, 9) (5, -7) (1, 4)

Multiple Choice

• 6

• -6

• 14

• -14

Multiple Choice

• 16

• 48

• -28

• 4

Multiple Choice

• -67

• 67

• 77

• -77

Multiple Choice

• -12

• -43

• 29

• -29

Multiple Choice

• -20

• 20

• -24

• 24

Multiple Choice

• 1

• -1

• 19

• -19

Multiple Choice

• -10

• -40

• 10

• 40

Multiple Choice

Find f(2)

• 5

• 4

• 3

• 0

Multiple Choice

Find f(3)

• 5

• 4

• 3

• 0

Multiple Choice

Find f(4)

• 5

• 4

• 3

• 0

Multiple Choice

Find f(-1)

• -3

• 15

• 19

• 17

Multiple Choice

Find f(-3)

• -3

• 15

• 19

• 17

Multiple Choice

If f(x) = 19, Find the value for x.

• 0

• 1

• 2

• 3

Multiple Choice

If f(x) = 4, Find the value for x.

• 0

• 1

• 2

• 3

Multiple Choice

Find f(1)

• 1

• 2

• -1

• 3

Multiple Choice

Find f(0)

• 1

• 2

• -1

• 3

Multiple Choice

If f(x) = 3 Find the value of x.

• 2

• 4

• 6

• 0

Multiple Choice

Find f(2)

• 1

• 4

• 2

• 0

Multiple Choice

Find f(2)

• 1

• 4

• 2

• 0

Multiple Choice

Find f(6)

• 1

• 4

• 2

• 0

Multiple Choice

Is this function linear or nonlinear?

• Linear

• Nonlinear

Multiple Choice

Is the function represented by this graph linear or nonlinear?

• Linear

• Nonlinear

Multiple Choice

Is the function represented by this table linear or nonlinear?

• Linear

• Nonlinear

Multiple Choice

Is the function $2x+3y=7\$  linear or nonlinear?

• Linear

• Nonlinear

Multiple Choice

Is the function  $y+x=4x-5$  linear or nonlinear?

• Linear

• Nonlinear

Multiple Choice

Is the function  $y=\frac{8}{x^2}$  linear or nonlinear?

• Linear

• Nonlinear

Multiple Choice

Is this relation a function?

• Yes, it is a function.

• No, it is not a function.

Multiple Choice

This table shows the height of my green bean plants that I planted a few weeks ago. Is this a linear or nonlinear function?

• Linear

• Nonlinear

Multiple Choice

Which statement best describes this graph?

• Linear function

• Nonlinear function

• Not a function

Multiple Choice

Is the graph pictured a linear or nonlinear function?

• Linear

• Nonlinear

Multiple Choice

Is this function linear or nonlinear?

• Linear

• Nonlinear

Multiple Choice

Is this function linear or nonlinear?

• linear function

• nonlinear function

Multiple Choice

Is this function linear or nonlinear?

• Linear function

• Nonlinear function

Multiple Choice

Which graph is linear?

• Graph A

• Graph B

• Graph C

• Graph D

Multiple Choice

Which equation represents y as a linear function of x?

• y=8x

• x=5

• y=4x²

• y=x³

Multiple Select

Which equations describe a nonlinear function? Select all that apply.

• x - 3y=12

• y=x³

• 3y=x

• y=5x²+6x

• 4x+3y²=24

Multiple Choice

To identify non-linear equations, we look for...

• exponents

• parentheses

Multiple Select

Which show a linear function? Select 2.

Multiple Choice

Select the linear equation.

Multiple Choice

Select the linear equation.

Multiple Choice

Is this function linear or nonlinear?

• Linear function

• Nonlinear function

Multiple Choice

Is this function linear or nonlinear?

• Linear function

• Nonlinear function

Multiple Choice

Is the representation linear or non-linear? 

• Linear
• Non-Linear

Multiple Choice

Is the representation linear or non-linear? 

• Linear
• Non-Linear