\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{\mathrm{sec} } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{\mathrm{arccot} } \) \( \newcommand{\arcsec}{\mathrm{arcsec} } \) \( \newcommand{\arccsc}{\mathrm{arccsc} } \) \( \newcommand{\sech}{\mathrm{sech} } \) \( \newcommand{\csch}{\mathrm{csch} } \) \( \newcommand{\arcsinh}{\mathrm{arcsinh} } \) \( \newcommand{\arccosh}{\mathrm{arccosh} } \) \( \newcommand{\arctanh}{\mathrm{arctanh} } \) \( \newcommand{\arccoth}{\mathrm{arccoth} } \) \( \newcommand{\arcsech}{\mathrm{arcsech} } \) \( \newcommand{\arccsch}{\mathrm{arccsch} } \)

17Calculus Precalculus - Domain and Range

Algebra

Polynomials

Functions

Rational Functions

Graphing

Matrices

Systems

Trigonometry

Complex Numbers

Applications

Tools

Articles

The domain is the set of all allowable values that a function can accept as input and produce a meaningful value.
The range is the set of all meaningful values that come out of a function.
Note - Discussion on the domain of composite functions can be found on the composite functions page.

Why Domain Is Important In Calculus

A function cannot be defined without a domain. But wait! Teachers (and textbooks) do it all the time, right?! Well . . . it may seem like it, but that's not really what is going on. Most teachers (and textbooks) don't come right out and say what the domain is. However, if they don't, that just means that the domain is implied by the equation. (One of the most common things you will see in math textbooks and classes is not stating what the authors and teachers think is obvious that may not be obvious to you. One of your jobs while learning math is to fill in the blanks.)

Here is why domain is important in calculus: Everything in calculus requires the understanding of the domain since the domain (and sometimes the range too) will often determine if and when you can apply a theorem or use a technique. So to be able to do calculus problems, you absolutely need to know the domain. Every fully-defined function includes domain, whether the domain is stated or not.

Most of the time the domain is implied and not explicitly stated. However, don't let that fool you. Important: Two equations with different domains are considered different functions, even if one or both domains are not explictly stated. Keep this in mind when you are working with limits. Good teachers will emphasize this and require your notation to reflect this knowledge. Here is an example. Assuming we are working with only real numbers (this is true most of the time in calculus, unless otherwise stated or implied by the context), let's compare these two functions.

\(\displaystyle{ f(x) = \frac{(x+1)(x-1)}{(x+1)} }\)

 

\(\displaystyle{ g(x) = x-1 }\)

If you look closely, you can see that you can cancel \(x+1\) in \(f(x)\) to get \(g(x)\). However, \(f(x) \neq g(x)\). Why? Because the domains are different. The domain of \( f(x) \) is all real numbers except for \(x = -1\) The domain of \(g(x)\) is all real numbers. In calculus, you can never just say \( f(x) = g(x)\) for these two functions.

However, you CAN say that \( f(x) = g(x) \) IF you also say \(x \neq -1\), but this must be explicitly stated since it is not obvious for \(g(x)\). So be very careful.

Limits - - Another thing you can say about these two functions is \(\displaystyle{ \lim_{x \rightarrow -1}{f(x)} = \lim_{x \rightarrow -1}{g(x)} }\) and you don't need to say anything about what is going on at \(x = -1\). This is true because of the limit key explained on the 17calculus limits page.
However, in this equation we are talking about the LIMITS being equal, not the functions themselves being equal, an important distinction.

What Are The Domain and Range?

To get started, we need to understand what the domain and range are. Most students think they know but take a few minutes to watch this video to make sure you really DO know. Then you can move forward with more confidence.

Dr Chris Tisdell - Domain and range of a function [13min-59secs]

video by Dr Chris Tisdell

For example, if the function is \( f(x) = \ln(x) \) and we want only real numbers to come out, we can't put in zero or any negative number for \(x\). Therefore, the domain is \( \{ x ~|~ x > 0 \} \). Another example is the square root function. We are allowed to take square roots of positive numbers and zero but, for calculus (since we usually work only with real numbers), we do not take square roots of negative numbers. So the domain of the square root function is \( \{ x ~|~ x \geq 0 \} \).

Finding Domain and Range From A Graph

This is the easiest case since a graph will tell us a lot about a function. For the domain, we just scan left-to-right and see if the graph is above, below or touching the x-axis, in which case, the x-values are in the domain. Similarly for the range, we scan vertically to see if the graph is to the left, right or touching the y-axis and, if so, then the y-values are in the range. Here are some practice problems to help you understand this better.

Instructions - - Unless otherwise instructed, find the domain and the range from the graphs.

Problem Statement

Solution

427 video

video by PatrickJMT

close solution

Problem Statement

Solution

428 video

video by PatrickJMT

close solution

Problem Statement

Solution

429 video

video by PatrickJMT

close solution

Problem Statement

Solution

430 video

video by PatrickJMT

close solution

Problem Statement

Solution

The graph given in the problem statement is slightly different than she has in the video. However, the domain and range are the same in both cases. Only the values of f(1) and f(5) differ. But you were not asked to find those in the problem statement.

432 video

video by Krista King Math

close solution

Problem Statement

Solution

From the graph, you can see that there are asymptotes at \(x=-1\) and \(x=2\). All other x-values will work. So the domain is all real numbers except for -1 and 2. The range is all real numbers.

close solution

Problem Statement

Solution

431 video

video by PatrickJMT

close solution

Problem Statement

Solution

From the graph, you can see that there are vertical asymptotes at \(x=-1\) and \(x=2\). Also notice there are no x-values in the interval \(-1 < x < 2\). So the domain is \(\{ x ~|~ x < -1 \cup x > 2 \}\).
The range is a bit more challenging. At the vertical asymptotes, the graph goes off to infinity in both the positive and negative direction. However, around \(y=1\), it is not clear what is going on. You should have enough experience with horizontal asymptotes to be able to recognize that there is a horizontal asymptote at \(y=1\). So the range is all real numbers except for 1.

close solution

Finding Domain and Range From The Equation

Let's start by discussing how to determine the domain of a function from the equation. (Finding it from the graph is easier but in calculus you need to be able to work only from the equation.) You need to know the domain before you can find the range. The domain is a list of \(x\)-values that are allowed to be put into the function. As you should already know, some functions are just not defined for certain values. When you try to put those values into a function, either nothing comes out or what does, doesn't make sense.

Steps To Determine Domain

Rather than trying to figure out what WILL work, it is much easier to determine what you can't put into a function. So, to determine the domain, we start with the set of all real numbers (in calculus, we usually stay with real numbers unless explicitly stated otherwise) and then remove the numbers that don't work. What is left is the domain. Here are the cases you will come across most of the time of situations that you don't want to have as input values.
1. In rational expressions, you get zero in the denominator.
2. In even roots, you have negative numbers (zero is okay).
3. In logarithms, you have numbers less than or equal to zero.
4. In the inverse trig functions arcsine and arccosine, values greater than 1 or less than -1.

The idea is to start out with all real numbers and remove numbers that fall under one of the above cases. What remains is the domain. Here are a few important things to remember.
- The domain of polynomials is the set of all real numbers.
- When combining terms, find the domain of each piece and then take the intersection of the sets to get the domain of the entire expression.
Here are a few functions and their domains. Knowing these should be enough for you to work most calculus problems.

Finding The Range

Once you have the domain, you can find the range. The range is not as easily determined. I usually use a combination of the domain, the graph and these simple rules.
1. Even roots are always positive.
2. Logarithms can have both positive and negative output.

function type

domain

polynomials

all real numbers

rational functions

all real numbers except where the denominator is zero

root functions

even roots: real numbers greater than or equal to zero;
odd roots: all real numbers

trig functions

sine and cosine: all real numbers
other trig functions: all real numbers except for angles that would produce a zero in the denominator

exponential functions

all real numbers

logarithm functions

real numbers greater than zero (not including zero)

Okay, try out these ideas on these practice problems.

Instructions - - Unless otherwise instructed, find the domain and the range of these functions.

\(\displaystyle{f(x)=\frac{1}{x-2}}\)

Problem Statement

\(\displaystyle{f(x)=\frac{1}{x-2}}\)

Solution

433 video

video by PatrickJMT

close solution
\(\displaystyle{f(x)=\frac{1}{x^2-x-6}}\)

Problem Statement

\(\displaystyle{f(x)=\frac{1}{x^2-x-6}}\)

Solution

434 video

video by PatrickJMT

close solution
\(f(x)= \sqrt{2x-8}\)

Problem Statement

\(f(x)= \sqrt{2x-8}\)

Solution

445 video

video by Khan Academy

close solution
\(\displaystyle{f(x)=\frac{\sqrt{x-1}}{x^2+4}}\)

Problem Statement

\(\displaystyle{f(x)=\frac{\sqrt{x-1}}{x^2+4}}\)

Solution

435 video

video by PatrickJMT

close solution
\(\displaystyle{f(x)=\ln(x-8)}\)

Problem Statement

\(\displaystyle{f(x)=\ln(x-8)}\)

Solution

437 video

video by PatrickJMT

close solution
Find the domain and range of the function \(f(x)=10^{2x}+\ln(21-3x)\).

Problem Statement

Find the domain and range of the function \(f(x)=10^{2x}+\ln(21-3x)\).

Final Answer

domain: \(x < 7\); range: \(y > 3.17771\)

Problem Statement

Find the domain and range of the function \(f(x)=10^{2x}+\ln(21-3x)\).

Solution

For the range in this problem, the presenter states that the range is \(y > 0\). However, if you graph the function, as we have done below, you can see that the range is actually closer to \(y > 3\). With the training you have in precalculus, this is pretty much as close as you can get using equations and looking at the graph. However, if you graph the function and then have the calculator find the minimum, it occurs at \(x \approx -0.77703\) giving the range closer to \(y > 3.17771\) (approximately).

1908

1908 video

video by Krista King Math

Final Answer

domain: \(x < 7\); range: \(y > 3.17771\)

close solution
For \(f(x)=x^3+2x^2\) and \(g(x)=3x^2-1\), find the domains of \((f+g)(x),\) \((f-g)(x), (f\cdot g)(x), (f/g)(x)\).

Problem Statement

For \(f(x)=x^3+2x^2\) and \(g(x)=3x^2-1\), find the domains of \((f+g)(x),\) \((f-g)(x), (f\cdot g)(x), (f/g)(x)\).

Solution

443 video

video by Krista King Math

close solution
\(\displaystyle{f(x)=\frac{1}{\sqrt{x^2-4}}}\)

Problem Statement

\(\displaystyle{f(x)=\frac{1}{\sqrt{x^2-4}}}\)

Solution

436 video

video by PatrickJMT

close solution

domain and range 17calculus youtube playlist

Here is a playlist of the videos on this page.

Really UNDERSTAND Precalculus

To bookmark this page and practice problems, log in to your account or set up a free account.

Calculus Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations Topics Listed Alphabetically

Precalculus Topics Listed Alphabetically

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

effective study techniques

Get great tutoring at an affordable price with Chegg. Subscribe today and get your 1st 30 minutes Free!

The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2020 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics
17Calculus
We use cookies to ensure that we give you the best experience on our website. By using this site, you agree to our Website Privacy Policy.