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Domain and Range of Functions

Author: Sophia
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what's covered
In this lesson, you will explore the domain and range of a function. This is very important since the domain and range collectively tell us the possible inputs and outputs of a function. Specifically, this lesson will cover:

Table of Contents

1. Finding the Domain of a Function Defined by an Equation

First, recall the definitions of domain and range. The domain of a function is the set of all possible values that can be used as inputs to a function. The range of a function is the set of all possible outputs that can be obtained from a function.

First, we’ll focus on finding the domain of a function when given its equation. This is mostly straightforward for functions having certain forms.

Given an equation of a function, the domain can be determined by a variety of factors.

  • Does the function have a variable in the denominator? If so, exclude any values for which the denominator is equal to 0.
  • Does the function involve an even root (square root, 4th root, etc.)? If so, only include values for which the radicand is nonnegative.
  • If there are no restrictions on the input value, then assume the domain is the set of all real numbers.

EXAMPLE

Find the domain of the function f open parentheses x close parentheses equals 3 x plus 5.

Things to note:
  • f open parentheses x close parentheses contains no fraction with a variable denominator.
  • f open parentheses x close parentheses contains no radicals.
There are no restrictions on what numbers can be substituted for f open parentheses x close parentheses. Thus, the domain of f open parentheses x close parentheses is the set of all real numbers. Using interval notation, this is written as open parentheses short dash infinity comma space infinity close parentheses.

hint
In this course and in calculus, interval notation is usually the preferred way to express domains and ranges of functions. In all future examples, domains will be expressed using interval notation.

try it
Consider the function f open parentheses x close parentheses equals x cubed minus 4 x plus 2.
Determine the domain of f   (x  ) and express it using interval notation.
open parentheses short dash infinity comma space infinity close parentheses

EXAMPLE

Find the domain of the function f open parentheses x close parentheses equals fraction numerator 2 x over denominator x squared minus 4 end fraction.

The function has a variable denominator. This means that f open parentheses x close parentheses is undefined when x squared minus 4 equals 0. Solve this equation to find the specific values of x.

table attributes columnalign left end attributes row cell space space space space space space space space space space space x squared minus 4 equals 0 end cell row cell open parentheses x plus 2 close parentheses open parentheses x minus 2 close parentheses equals 0 end cell row cell space space space space space space space space space space space space space x plus 2 equals 0 space or space x minus 2 equals 0 end cell row cell space space space space space space space space space space space space space space space space space space space x equals short dash 2 space or space x equals 2 end cell end table

This means that f open parentheses x close parentheses is undefined when x equals 2 or when x equals short dash 2.

Thus, the domain is the set of all real numbers except plus-or-minus 2. This could be written as x not equal to plus-or-minus 2. Using interval notation, the domain is written as open parentheses short dash infinity comma space short dash 2 close parentheses union open parentheses short dash 2 comma space 2 close parentheses union open parentheses 2 comma space infinity close parentheses.

try it
Consider the function f open parentheses x close parentheses equals fraction numerator 3 x plus 2 over denominator 4 x minus 1 end fraction.
Determine the domain of f   (x  ) and express it using interval notation.
The function f open parentheses x close parentheses is undefined for any values of x where the denominator is equal to 0. Setting 4 x minus 1 equals 0 gives x equals 1 fourth.

This means that the domain of f open parentheses x close parentheses is the set of all real numbers, excluding 1 fourth.

Written using interval notation, the domain of f open parentheses x close parentheses is open parentheses short dash infinity comma space 1 fourth close parentheses union open parentheses 1 fourth comma space infinity close parentheses.

Now we’ll look at functions that contain radicals with even roots (square roots, 4th roots, etc.).

EXAMPLE

Find the domain of the function f open parentheses x close parentheses equals 1 plus square root of 2 x minus 3 end root. Express the domain using interval notation.

When only real numbers are considered, recall that only square roots of nonnegative numbers are defined. This means that the domain of this function is the set of values of x for which 2 x minus 3 greater or equal than 0.

Now, solve the inequality:

table attributes columnalign left end attributes row cell 2 x minus 3 greater or equal than 0 end cell row cell space space space space space space 2 x greater or equal than 3 end cell row cell space space space space space space space space x greater or equal than 3 over 2 end cell end table
Using interval notation, the inequality x greater or equal than 3 over 2 is written as open square brackets 3 over 2 comma space infinity close parentheses. Thus, the domain of the function is open square brackets 3 over 2 comma space infinity close parentheses.

try it
Consider the function f open parentheses x close parentheses equals fourth root of 5 minus x end root plus 3.
Determine the domain of f   (x  ) and express it using interval notation.
Since f open parentheses x close parentheses is a radical function with an even root, the domain is the set of all real numbers for which the radicand is nonnegative.

Using that logic, set 5 minus x greater or equal than 0.

Adding x to both sides, we have 5 greater or equal than x comma or x less or equal than 5.

Written using interval notation, the domain of f open parentheses x close parentheses is open parentheses short dash infinity comma space 5 close square brackets.

Since we just looked at functions that contain radicals with even roots, we’ll now examine functions that contain radicals with odd roots.

EXAMPLE

Find the domain of the function f open parentheses x close parentheses equals cube root of x plus 2 end root. Express the domain using interval notation.

Recall that the cube root (or any odd root of a number) can be applied to any real number (positive, zero, or negative). Since the radicand has no restrictions on it as well, the domain of f open parentheses x close parentheses is the set of all real numbers. Using interval notation, this is expressed as open parentheses short dash infinity comma space infinity close parentheses.

try it
Consider the function f open parentheses x close parentheses equals fifth root of x plus 4 end root.
Determine the domain of f   (x  ) and express it using interval notation.
Since f open parentheses x close parentheses is a radical function with an odd root, and the radicand, x plus 4 comma is never undefined, the domain of f open parentheses x close parentheses is the set of all real numbers. Using interval notation, this is expressed as open parentheses short dash infinity comma space infinity close parentheses.

Here is an example that combines a few ideas.

EXAMPLE

Determine the domain of the function f open parentheses x close parentheses equals fraction numerator 2 over denominator square root of x minus 3 end root end fraction.

First, notice that the function contains a square root. The square root is defined for values of x where the radicand is nonnegative:

table attributes columnalign left end attributes row cell x minus 3 greater or equal than 0 end cell row cell space space space space space space x greater or equal than 3 end cell end table

Also, the radical is in the denominator. This means that the radicand cannot equal 0. This means that x not equal to 3.

Pulling these together, the domain of the function is the set of all values for which x greater or equal than 3 and x not equal to 3. More simply stated, the domain is x greater than 3.

Written using interval notation, the domain is expressed as open parentheses 3 comma space infinity close parentheses.

It is not always convenient to find the domain of a function given only its equation. The graph of the function can be very useful in determining both the domain and the range of a function.

terms to know
Domain
The set of all possible values that can be used as inputs to a function.
Range
The set of all possible outputs that can be obtained from a function.

2. Finding Domain and Range From Graphs

Recall that the graph of a function is a visual representation of all solutions to an equation. Recall also that the domain is the set of all possible inputs of a function. On a graph, these are represented as values on the x-axis. Likewise, the range is the set of all output values of a function, which are shown as values on the y-axis.

For instance, consider the graph shown below.

  • Looking at the graph from left to right, the smallest x-value that is represented on the graph is -5, and the graph appears to continue to the right indefinitely. Thus, the domain of the function is open square brackets short dash 5 comma space infinity close parentheses.
  • Observe that the highest point on the graph is open parentheses short dash 5 comma space 5 close parentheses. Looking at the graph vertically, all y-values that are lower than 5 are also represented on this graph. Thus, the range of the function is open parentheses short dash infinity comma space 5 close square brackets.

EXAMPLE

Determine the domain and range of the function whose graph is shown below.



Looking at the graph from left to right, notice that the leftmost point, open parentheses short dash 3 comma space 0 close parentheses comma is an open circle. This means that x equals short dash 3 is not included. The rightmost point on the graph is open parentheses 1 comma space short dash 4 close parentheses comma which is solid, meaning it is included. The domain of this function is open parentheses short dash 3 comma space 1 close square brackets.

Looking at the graph vertically, the lowest y-value represented on the graph is -4 (located at open parentheses short dash 2 comma space 4 close parentheses and open parentheses 1 comma space short dash 4 close parentheses), while the highest y-value is 0 (located at open parentheses 0 comma space 0 close parentheses). The range of this function is open square brackets short dash 4 comma space 0 close square brackets.

try it
Determine the domain and range of the function whose graph is shown below:



Determine the domain.
open square brackets short dash 4 comma space infinity close parentheses
Determine the range.
open square brackets 0 comma space infinity close parentheses

big idea
In manufacturing, suppose the total cost of producing x pounds of a certain material is estimated by the function C open parentheses x close parentheses equals 12 x plus 400. From what we know about the domain already, it should be the set of all real numbers. However, in this situation, it makes no sense for x to be a negative number, since x is the number of pounds. Therefore, a more sensible domain is open square brackets 0 comma space infinity close parentheses.

The moral of the story is that the domain is determined by two things: the function itself, and the context of the function (if any is given).

summary
In this lesson, you recalled that the domain and range provide useful information about a function in that they tell us the sets of numbers that can be used for inputs and outputs, respectively. You learned how to find the domain of a function defined by an equation using interval notation, determined by a variety of factors such as whether the function has a variable in the denominator or involves an even index root, and if there are no restrictions on the input value. Recalling that the graph of a function is a visual representation of all solutions to an equation, you also learned how to find domain and range from graphs, where the domain is represented as values on the x-axis as you scan from left to right, and the range is shown as values on the y-axis as you scan from bottom to top. Keep in mind that special care needs to be taken in real-life situations where certain values of x (or y) do not make sense in the situation (for instance, a negative length, or a fractional number of items).

SOURCE: THIS WORK IS ADAPTED FROM CHAPTER 0 OF CONTEMPORARY CALCULUS BY DALE HOFFMAN AND PRECALCULUS BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/BOOKS/PRECALCULUS/PAGES/1-INTRODUCTION-TO-FUNCTIONS

Terms to Know
Domain

The set of all possible values that can be used as inputs to a function.

Range

The set of all possible outputs that can be obtained from a function.

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