Topics
4. Functions & Linear Relationships
topic badge
United States of AmericaTennessee
Algebra 1

4.02 Domain and range

Lesson
Practice
Lesson

Concept summary

Two defining parts of any function are its domain and range.

A curved function graphed on a coordinate plane with an unfilled endpoint at the left most part of the function and a filled endpoint at the right most part of the function. A segment across the top of the function marks the domain with an unfilled enpoint to the left and a filled endpoint to the right

The set of all possible input values (x-values) for a function or relation is called the domain.

In the example shown, the domain is the interval of values -3 < x \leq 1. Notice that -3 is not included in the domain, which is indicated by the unfilled point.

A curved function graphed on a coordinate plane with an unfilled endpoint at the left most part of the function and a filled endpoint at the right most part of the function. A vertical segment to the right of the function marks the range with filled endpoints at the top and bottom

The set of all possible output values (y-values) for a function or relation is called the range.

In the example shown, the range is the interval of values -4 \leq y \leq 0. Notice that both endpoints are included in the range, since the function reaches a height of y = 0 at the origin.

A domain which is made up of disconnected values is said to be a discrete domain.

A four quadrant coordinate plane with seven points plotted in different quadrants. The points are located at (negative 3, 0), ( negative 54, 3),(negative 2.5, negative 1), (0, negative 3), (1.5, negtaive 2.5), (3, 0) and (3.5, 2).

A function with a discrete domain. It is only defined for distinct x-values.

A domain made up of a single connected interval of values is said to be a continuous domain.

A four quadrant coordinate plane plotted with a parabola, a continuous curve that opens upward. The parabola has its vertex at (0, negative 3), and passes through (negative 3, 0) and (3, 0)

A function with a continuous domain. It is defined for every x-value in an interval.

It is possible for the domain of a function to be neither discrete nor continuous. An example of this is a step function.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

A step function is made up of pieces that are horizontal line segments or points.

A function cannot have two different y-values for the same x-value, so we use filled and unfilled points to show which step the y-value is on.

The domain and range of a function are commonly expressed using inequality notation or set-builder notation.

Note that if two functions have different domains, then they must be different functions, even if they take the same values on the shared parts of their domains.

Worked examples

Example 1

Consider the function shown in the graph.

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
a

State whether the function has a discrete or continuous domain.

Solution

The function is defined at every value of x across an interval, so it has a continuous domain.

b

Determine the domain of the function using set-builder notation.

Solution

We can see that the function is defined for every x-value between -6 and 8, including -6 but not including 8.

So the domain of the function can be written as \text{Domain: } \left\{x\, \vert\, -6 \leq x < 8\right\}

c

Determine the range of the function using set-builder notation.

Solution

We can see that the function reaches every y-value between -6 and 4, including 4 but not including -6.

So the range of the function can be written as \text{Range: } \left\{y\, \vert\, -6 < y \leq 4\right\}

Example 2

Consider the following step function.

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
a

Determine the domain of the function using set-builder notation.

Solution

We can see that the function is made up of three pieces.

The piece furthest to the left is defined for every x-value between -8 and -5, inluding -8 but not including -5.

The piece in the middle is defined for every x-value between -4 and -2, not inluding -4 or -2.

The piece furthest to the right is defined for every x-value between -2 and 6, including both points.

Because the pieces in the middle and furthest to the right have an overlap, at x=-2, we can have a single domain for the two pieces.

So the domain of the function can be written as\text{Domain: } \left\{x: \begin{aligned}-8&\leq x < -5 \\-4&\leq x \leq 6 \end{aligned}\right\}

b

Determine the range of the function using set-builder notation.

Solution

The piece furthest to the left is defined for every x-value between -8 and -5 at y=6.

The piece in the middle is defined for every x-value between -4 and -2 at y=2.

The piece furthest to the right is defined for every x-value between -2 and 6 at y=-4.

So the range of the function can be written as\text{Range: } \{y= 6, 2, -4\}

Or rearranging the values from least to greatest \text{Range: } \{y= -4, 2, 6\}

Outcomes

A1.N.Q.A.1

Use units as a way to understand real-world problems.*

A1.N.Q.A.1.C

Define and justify appropriate quantities within a context for the purpose of modeling.*

A1.F.IF.A.1

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

A1.F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the context of the function it models. *

A1.MP1

Make sense of problems and persevere in solving them.

A1.MP2

Reason abstractly and quantitatively.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP4

Model with mathematics.

A1.MP6

Attend to precision.

A1.MP8

Look for and express regularity in repeated reasoning.

What is Mathspace

About Mathspace