Explainer: Domain and Range of a Function

In this explainer, we will learn how to identify the domain and range of functions from their equations.

Recall that a function ๐‘“ assigns to members of a one set, say ๐‘‹, values in a second set, ๐‘Œ. We think of ๐‘‹ as the source of inputs to the function and ๐‘Œ as the target of its outputs. So, for an input ๐‘ฅ from ๐‘‹, the output given by ๐‘“ is the value ๐‘“(๐‘ฅ)โˆˆ๐‘Œ.

Definitions: Domain and Range

  1. The domain of ๐‘“ is the set of inputs ๐‘ฅโˆˆ๐‘‹ for which ๐‘“(๐‘ฅ) is defined.
  2. The range of ๐‘“ is the set of all outputs ๐‘“(๐‘ฅ) as ๐‘ฅ wanders over the domain.

Notice that the range is a subset of ๐‘Œ.

Generally, the domain is all of ๐‘‹, particularly when the function is given as a set of ordered pairs or by a diagram.

For example, the following defines a function into the set ๐‘Œ={0,1,2,3,4}: ๐‘”={(๐‘Ž,1),(๐‘,1),(๐‘,2),(๐‘‘,4)}.

  • The domain of ๐‘” is the set {๐‘Ž,๐‘,๐‘,๐‘‘}.
  • The range of ๐‘” is the set of all values taken by ๐‘”, and is therefore {๐‘”(๐‘Ž),๐‘”(๐‘),๐‘”(๐‘),๐‘”(๐‘‘)}={1,1,2,4}={1,2,4}, a proper subset of ๐‘Œ.

The instances where the domain needs to be identified are usually when the function is defined for subsets of the real numbers. So, you will meet questions like, โ€œWhat is the domain of โ„Ž(๐‘ฅ)=1๐‘ฅ?โ€

What this question is asking is, โ€œWhat is the largest subset of the real numbers โ„ for which the assignment ๐‘ฅโ†ฆ1๐‘ฅ makes sense?โ€ The solution, of course, is all the nonzero numbers, the set โ„โˆ’{0}. Or, put another way, the set {๐‘ โˆˆโ„โˆฃ๐‘ โ‰ 0}.

Sometimes, domains for functions between sets of real numbers must be determined from a graph, such as the following example.

Example 1: Determining the Domain of a Function from Its Graph

Determine the domain of the function represented in the shown graph.

Answer

The solid dot at the extreme left of the curve tells us that ๐‘“(4)=1, and the arrow at the other end says that the function is defined for real numbers to the right of the axis. Therefore, the domain of this function ๐‘“ is {๐‘ฅโˆˆโ„โˆฃ๐‘ฅโ‰ฅ4}=[4,โˆž).

More often, determining the domain of a function defined by a formula involves deciding what the permitted inputs are.

Example 2: Determining the Domain of a Radical Function

Determine the domain of the function ๐‘“(๐‘ฅ)=โˆš|๐‘ฅ|โˆ’33.

Answer

The square root is only valid for nonnegative numbers, so we require |๐‘ฅ|โˆ’33โ‰ฅ0 or |๐‘ฅ|โ‰ฅ0.

If ๐‘ฅโ‰ฅ0, this means ๐‘ฅ=|๐‘ฅ|โ‰ฅ33 (i.e., ๐‘ฅโ‰ฅ33).

If ๐‘ฅ<0, then, by definition, โˆ’๐‘ฅ=|๐‘ฅ|โ‰ฅ33, which is the same as ๐‘ฅโ‰คโˆ’33.

So, our domain is the set {๐‘ฅโˆˆโ„โˆฃ๐‘ฅโ‰ฅ33๐‘ฅโ‰คโˆ’33}={๐‘ฅโˆˆโ„โˆฃ๐‘ฅโ‰ฅ33}โˆช{๐‘ฅโˆˆโ„๐‘ฅโ‰คโˆ’33}=(โˆ’โˆž,โˆ’33]โˆช[33,โˆž).or

Of course, this is not an interval, although you can write it as the complement of one: โ„โˆ’(โˆ’33,33).

Here is another example.

Example 3: Finding the Domain of a Piecewise Defined Function

Determine the domain of the function ๐‘“(๐‘ฅ)=๏ฎ๐‘ฅ+4๐‘ฅโˆˆ[โˆ’4,4],โˆ’8๐‘ฅ+40๐‘ฅโˆˆ(4,5].ifif

Answer

In this case, the figure is not even necessary to answer the question. ๐‘ฅ lies in the domain, provided that it satisfies one of the conditions of the piecewise definition. This means that ๐‘ฅ belongs to the union [โˆ’4,4]โˆช(4,5], which we see is actually the same as [โˆ’4,5].

Suppose that ๐‘“ is a function on a domain ๐‘‹. By definition, the range of ๐‘“ is the set of all its outputs: {๐‘“(๐‘ฅ)โˆฃ๐‘ฅโˆˆ๐‘‹}. If ๐‘‹ is part of the real line and ๐‘“ is represented by a graph, this range is visible as part of the ๐‘ฆ-axis.

Example 4: Finding the Range of a Quadratic Function from Its Graph

Determine the range of the function represented by the shown graph.

Answer

We can see that if this is the function ๐‘“, then ๐‘“(โˆ’7)=1, and this is the maximum value of the function. There is no ๐‘ฅ with ๐‘“(๐‘ฅ)>1. At the same time, the graph appears to be a parabola that takes on all values less than or equal to 1. The range is {๐‘ฆโˆˆโ„โˆฃ๐‘ฆโ‰ค1}=(โˆ’โˆž,1].

Even when we are not supplied with a graph, if we have a formula describing a function, we can determine the range from that. For example, with ๐‘“(๐‘ฅ)=โˆš|๐‘ฅ|โˆ’33 above, we saw that the numbers ๐‘ฅ=ยฑ33 were important for the domain. So, ๐‘“(33)=โˆš|33|โˆ’33=โˆš33โˆ’33=โˆš0=0. Similarly, ๐‘“(โˆ’33)=0.

We guess that the range of ๐‘“ is just all nonnegative numbers, the interval [0,โˆž). This is the claim that any ๐‘ฆโ‰ฅ0๐‘“(๐‘ฅ)is for some ๐‘ฅ. For example, suppose we took ๐‘ฆ=1,234, what is ๐‘ฅ so that ๐‘“(๐‘ฅ)=1,234?

In other words, what ๐‘ฅ makes โˆš|๐‘ฅ|โˆ’33=1,234?

The answer is found by solving the equation. We need |๐‘ฅ|โˆ’33=1,234=1,522,756๏Šจ. So, |๐‘ฅ|=1,522,789, and we have two values with target 1,234: ๐‘ฅ=1,522,789 and ๐‘ฅ=โˆ’1,522,789. It is easy to see why any ๐‘ฆโ‰ฅ0 will do.

Example 5: Finding the Range of a Linear Function from Its Domain

If ๐‘“โˆถ[2,21]โ†’โ„, where ๐‘“(๐‘ฅ)=3๐‘ฅโˆ’10, find the range of ๐‘“.

Answer

The graph of ๐‘“ is a straight line, so we know that for ๐‘ฅโˆˆ[2,21], the values ๐‘“(๐‘ฅ) all lie between the values at the endpoints, which are included. These values are ๐‘“(2)=3(2)โˆ’10=โˆ’4,๐‘“(21)=3(21)โˆ’10=53.

In other words, therangeofthesetofvaluessatisfying๐‘“=๐‘ฆโˆ’4โ‰ค๐‘ฆโ‰ค53=[โˆ’4,53].

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