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 𝑓 assigns to members of a one set, say 𝑋, values in a second set, π‘Œ. We think of 𝑋 as the source of inputs to the function and π‘Œ as the target of its outputs. So, for an input π‘₯ from 𝑋, the output given by 𝑓 is the value 𝑓(π‘₯)βˆˆπ‘Œ.

Definitions: Domain and Range

  1. The domain of 𝑓 is the set of inputs π‘₯βˆˆπ‘‹ for which 𝑓(π‘₯) is defined.
  2. The range of 𝑓 is the set of all outputs 𝑓(π‘₯) as π‘₯ wanders over the domain.

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

Generally, the domain is all of 𝑋, 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 π‘Œ={0,1,2,3,4}: 𝑔={(π‘Ž,1),(𝑏,1),(𝑐,2),(𝑑,4)}.

  • The domain of 𝑔 is the set {π‘Ž,𝑏,𝑐,𝑑}.
  • The range of 𝑔 is the set of all values taken by 𝑔, and is therefore {𝑔(π‘Ž),𝑔(𝑏),𝑔(𝑐),𝑔(𝑑)}={1,1,2,4}={1,2,4}, a proper subset of π‘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, β€œWhat is the domain of β„Ž(π‘₯)=1π‘₯?”

What this question is asking is, β€œWhat is the largest subset of the real numbers ℝ for which the assignment π‘₯↦1π‘₯ makes sense?” The solution, of course, is all the nonzero numbers, the set β„βˆ’{0}. Or, put another way, the set {π‘ βˆˆβ„βˆ£π‘ β‰ 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 𝑓(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 𝑓 is {π‘₯βˆˆβ„βˆ£π‘₯β‰₯4}=[4,∞).

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 𝑓(π‘₯)=√|π‘₯|βˆ’33.

Answer

The square root is only valid for nonnegative numbers, so we require |π‘₯|βˆ’33β‰₯0 or |π‘₯|β‰₯0.

If π‘₯β‰₯0, this means π‘₯=|π‘₯|β‰₯33 (i.e., π‘₯β‰₯33).

If π‘₯<0, then, by definition, βˆ’π‘₯=|π‘₯|β‰₯33, which is the same as π‘₯β‰€βˆ’33.

So, our domain is the set {π‘₯βˆˆβ„βˆ£π‘₯β‰₯33π‘₯β‰€βˆ’33}={π‘₯βˆˆβ„βˆ£π‘₯β‰₯33}βˆͺ{π‘₯βˆˆβ„π‘₯β‰€βˆ’33}=(βˆ’βˆž,βˆ’33]βˆͺ[33,∞).or

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

Suppose that 𝑓 is a function on a domain 𝑋. By definition, the range of 𝑓 is the set of all its outputs: {𝑓(π‘₯)∣π‘₯βˆˆπ‘‹}. If 𝑋 is part of the real line and 𝑓 is represented by a graph, this range is visible as part of the 𝑦-axis.

Example 3: 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 𝑓, then 𝑓(βˆ’7)=1, and this is the maximum value of the function. There is no π‘₯ with 𝑓(π‘₯)>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 {π‘¦βˆˆβ„βˆ£π‘¦β‰€1}=(βˆ’βˆž,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 𝑓(π‘₯)=√|π‘₯|βˆ’33 above, we saw that the numbers π‘₯=Β±33 were important for the domain. So, 𝑓(33)=√|33|βˆ’33=√33βˆ’33=√0=0. Similarly, 𝑓(βˆ’33)=0.

We guess that the range of 𝑓 is just all nonnegative numbers, the interval [0,∞). This is the claim that any 𝑦β‰₯0 is 𝑓(π‘₯) for some π‘₯. For example, suppose we took 𝑦=1,234, what is π‘₯ so that 𝑓(π‘₯)=1,234?

In other words, what π‘₯ makes √|π‘₯|βˆ’33=1,234?

The answer is found by solving the equation. We need |π‘₯|βˆ’33=1,234=1,522,756. So, |π‘₯|=1,522,789, and we have two values with target 1,234: π‘₯=1,522,789 and π‘₯=βˆ’1,522,789. It is easy to see why any 𝑦β‰₯0 will do.

Example 4: Finding the Range of a Linear Function from Its Domain

If π‘“βˆΆ[2,21]→ℝ, where 𝑓(π‘₯)=3π‘₯βˆ’10, find the range of 𝑓.

Answer

The graph of 𝑓 is a straight line, so we know that for π‘₯∈[2,21], the values 𝑓(π‘₯) all lie between the values at the endpoints, which are included. These values are 𝑓(2)=3(2)βˆ’10=βˆ’4,𝑓(21)=3(21)βˆ’10=53.

In other words, therangeofthesetofvaluessatisfying𝑓=π‘¦βˆ’4≀𝑦≀53=[βˆ’4,53].

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