Topics
5. Functions & Linear Relationships
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United States of AmericaTennessee
Integrated Math 1

5.02 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.

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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.

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

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.

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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

The function has no undefined points nor domain constraints. Therefore, the domain of the function can be written as\text{Domain: } \left\{x\, \vert\, -\infty \leq x <\infty\right\}

c

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

Solution

The range of an exponential function of the form c\times n^{ax+b}+k is f(x)>k and k=-2. So the range of the function can be written as \text{Range: } \left\{y\, \vert\,y > -2\right\}

Outcomes

M1.N.Q.A.1

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

M1.N.Q.A.1.C

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

M1.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).

M1.F.IF.B.5

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

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP6

Attend to precision.

M1.MP8

Look for and express regularity in repeated reasoning.

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