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

17Calculus Precalculus - Composite Functions

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

If you watched the function notation video on the main functions page, you got a taste of a concept called composite functions or composition of functions. The idea is that, instead of just plugging numbers in functions, you can also plug in more sophisticated things including other functions. That's basically all that is going on.

If you want a complete lecture on this topic, we recommend this video. He also gives you some information about operations on functions, which is good to review at this point.

Prof Leonard - Operations and Composition of Functions

video by Prof Leonard

Getting Started

Let's get started by looking at an example.
For the functions \(f(x)=3x^2\) and \(g(x)=x+3\), by now you know that \(f(3)=3(3)^2=27\) and \(g(2)=2+3=5\). But what is \(f(x+3)\)? Well, if you compare \(f(x+3)\) and \(f(x)\), you notice that we replaced \(x\) in \(f(x)\) with \(x+3\) to get \(f(x+3)\). So since we replace \(x\) with \(x+3\) in \(f(x)\), we do the same replacement in the function \(f(x)=3x^2\) to get \(f(x+3)=3(x+3)^2\).
Similarly, we can find \(g(3x^2)\) by replacing x with \(3x^2\) to get \(g(3x^2)=3x^2+3\).

Given the same two functions \(f(x)=3x^2\) and \(g(x)=x+3\), we can write the same two composition of functions as \(f(g(x))\) and \(g(f(x))\).

Here is a great video explaining composite functions with lots of examples.

MIP4U - Composite Functions [8min-55secs]

video by MIP4U

Another way to write the composition of the two functions \(f(x)\) and \(g(x)\) is \( f(g(x)) = (f \circ g)(x)\). You will also see this written as \(f \circ g\) without the \((x)\).
Be careful with composition, \(f(g(x)) \neq g(f(x))\) most of the time. There will be times that they are equal but those are very special cases (like with inverse functions).

Okay, time for some practice problems.

Schaum's Outline of Precalculus, 3rd Edition: 738 Solved Problems + 30 Videos

Practice

For \( f(x) = 3x - 1 \) and \( g(x) = 2x^2 + x + 1 \), find \( f \circ g \) and \( g \circ f \).

Problem Statement

For \( f(x) = 3x - 1 \) and \( g(x) = 2x^2 + x + 1 \), find \( f \circ g \) and \( g \circ f \).

Solution

2916 video solution

Log in to rate this practice problem and to see it's current rating.

For \( f(x) = 3x - 4 \) and \( g(x) = x^3 - 3 \), find \( f \circ g \) and \( g \circ f \).

Problem Statement

For \( f(x) = 3x - 4 \) and \( g(x) = x^3 - 3 \), find \( f \circ g \) and \( g \circ f \).

Solution

2917 video solution

Log in to rate this practice problem and to see it's current rating.

For \( f(x) = 5x + 2 \) and \( g(x) = x^3 - 4 \), calculate \( f(g(2)) \) and \( g(f(-1)) \).

Problem Statement

For \( f(x) = 5x + 2 \) and \( g(x) = x^3 - 4 \), calculate \( f(g(2)) \) and \( g(f(-1)) \).

Solution

2918 video solution

Log in to rate this practice problem and to see it's current rating.

For \( f(x) = 2x - 3 \) and \( g(x) = 5x + 1 \), find \( f \circ g \).

Problem Statement

For \( f(x) = 2x - 3 \) and \( g(x) = 5x + 1 \), find \( f \circ g \).

Solution

2919 video solution

Log in to rate this practice problem and to see it's current rating.

For \( f(x) = x + 3 \) and \( g(x) = x^2 - 5 \), find \( f \circ g \) and \( g \circ f \).

Problem Statement

For \( f(x) = x + 3 \) and \( g(x) = x^2 - 5 \), find \( f \circ g \) and \( g \circ f \).

Solution

2920 video solution

Log in to rate this practice problem and to see it's current rating.

For \( f(x) = 3x + 2 \) and \( g(x) = x^2 + 1 \), find \( f \circ g \) and \( g \circ f \).

Problem Statement

For \( f(x) = 3x + 2 \) and \( g(x) = x^2 + 1 \), find \( f \circ g \) and \( g \circ f \).

Solution

2921 video solution

Log in to rate this practice problem and to see it's current rating.

For \( f(x) = x^2 - 3 \) and \( g(x) = \sqrt{x-1} \), find \( f \circ g \) and \( g \circ f \).

Problem Statement

For \( f(x) = x^2 - 3 \) and \( g(x) = \sqrt{x-1} \), find \( f \circ g \) and \( g \circ f \).

Solution

2922 video solution

Log in to rate this practice problem and to see it's current rating.

For \(f(x)=x^2+x\), \(g(x)=4-x\), find \(f \circ g\) and \(g \circ f\)

Problem Statement

For \(f(x)=x^2+x\), \(g(x)=4-x\), find \(f \circ g\) and \(g \circ f\)

Solution

PatrickJMT - 1624 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

For \(f(x)=x^2-2x+3\), \(g(x)=2x+1\), find \(f \circ g\) and \(g \circ f\).

Problem Statement

For \(f(x)=x^2-2x+3\), \(g(x)=2x+1\), find \(f \circ g\) and \(g \circ f\).

Solution

MIP4U - 1626 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

\(f(x)=2x-3\), \(g(x)=6x^2-4x+5\), find \(f(g(7))\), \(g(f(2))\), \(f(g(-3))\), \(f(f(-4))\).

Problem Statement

\(f(x)=2x-3\), \(g(x)=6x^2-4x+5\), find \(f(g(7))\), \(g(f(2))\), \(f(g(-3))\), \(f(f(-4))\).

Solution

MIP4U - 1627 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

For \(f(x)=x^2+5\), \(g(x)=6x-15\), find \((f \circ g)(2)\) and \((g \circ f)(2)\).

Problem Statement

For \(f(x)=x^2+5\), \(g(x)=6x-15\), find \((f \circ g)(2)\) and \((g \circ f)(2)\).

Solution

MIP4U - 1628 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

For \(f(x)=3x/5+4\), \(g(x)=2x^2-5x+9\), find \((f \circ g)(2)\), \((g \circ f)(4)\), \((f \circ g)(1/2)\), \((f \circ f)(-4/5)\).

Problem Statement

For \(f(x)=3x/5+4\), \(g(x)=2x^2-5x+9\), find \((f \circ g)(2)\), \((g \circ f)(4)\), \((f \circ g)(1/2)\), \((f \circ f)(-4/5)\).

Solution

MIP4U - 1631 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

For \(f(x)=1/(x+5)\), \(g(x)=3/x-5\), find \(f \circ g\) and \(g \circ f\).

Problem Statement

For \(f(x)=1/(x+5)\), \(g(x)=3/x-5\), find \(f \circ g\) and \(g \circ f\).

Solution

MIP4U - 1629 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

For \(f(x)=x^3-x+1\), \(g(x)=2x^2\), \(h(x)=\sqrt{x}\), find \( f(g(h(x))) \).

Problem Statement

For \(f(x)=x^3-x+1\), \(g(x)=2x^2\), \(h(x)=\sqrt{x}\), find \( f(g(h(x))) \).

Solution

MIP4U - 1630 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

For \(f(x)=x/(x+1)\), \(g(x)=x^{10}\), \(h(x)=x+3\), find \(f \circ g \circ h\).

Problem Statement

For \(f(x)=x/(x+1)\), \(g(x)=x^{10}\), \(h(x)=x+3\), find \(f \circ g \circ h\).

Solution

Krista King Math - 1632 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

For \(h(x)=3x\), \(g(t)=-2t-2-h(t)\), \(f(n)=-5n^2+h(n)\), find \(h(g(8))\)

Problem Statement

For \(h(x)=3x\), \(g(t)=-2t-2-h(t)\), \(f(n)=-5n^2+h(n)\), find \(h(g(8))\)

Solution

Khan Academy - 1633 video solution

video by Khan Academy

Log in to rate this practice problem and to see it's current rating.

Domain of Composite Functions

To find the domain of composite functions, some care is required. Some instructors tell you to find the expression of the composite function and determine the domain from that. However, sometimes information is lost in the algebra and it is easy to get the incorrect answer.

The best way to determine the domain is to start with the inside function, determine the domain and range of it and then use it's range as the domain of the outside function.

Practice

For \( f(x) = 1/x \) and \( g(x) = x + 3 \), find \( (f \circ g)(x) \), \( (g \circ f)(x) \) and the domains of each composite function.

Problem Statement

For \( f(x) = 1/x \) and \( g(x) = x + 3 \), find \( (f \circ g)(x) \), \( (g \circ f)(x) \) and the domains of each composite function.

Solution

2923 video solution

Log in to rate this practice problem and to see it's current rating.

For \( f(x) = \sqrt{x-2} \) and \( g(x) = 1/(x+3) \), find \( f \circ g \) and it's domain.

Problem Statement

For \( f(x) = \sqrt{x-2} \) and \( g(x) = 1/(x+3) \), find \( f \circ g \) and it's domain.

Solution

2924 video solution

Log in to rate this practice problem and to see it's current rating.

For \( f(x) = \sqrt{x-5} \) and \( g(x) = 1/(8-x) \), find \( g \circ f \) and it's domain.

Problem Statement

For \( f(x) = \sqrt{x-5} \) and \( g(x) = 1/(8-x) \), find \( g \circ f \) and it's domain.

Solution

2925 video solution

Log in to rate this practice problem and to see it's current rating.

For \(f(x)=x+1/x\), \(g(x)=(x+1)/(x+2)\), find \(f \circ g\), \(g \circ f\), \(f \circ f\), \(g \circ g\) and their domains.

Problem Statement

For \(f(x)=x+1/x\), \(g(x)=(x+1)/(x+2)\), find \(f \circ g\), \(g \circ f\), \(f \circ f\), \(g \circ g\) and their domains.

Solution

Krista King Math - 1625 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

For \(\displaystyle{f(x)=x+\frac{1}{x}; }\) \(\displaystyle{ g(x)=\frac{x+1}{x+2}}\) find the domains of \(f\circ g,~~ g\circ f,~~f\circ f,~~ g\circ g\).

Problem Statement

For \(\displaystyle{f(x)=x+\frac{1}{x}; }\) \(\displaystyle{ g(x)=\frac{x+1}{x+2}}\) find the domains of \(f\circ g,~~ g\circ f,~~f\circ f,~~ g\circ g\).

Solution

Krista King Math - 444 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)= \frac{1}{x}; g(x)=\sqrt{x+4}}\).

Problem Statement

Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)= \frac{1}{x}; g(x)=\sqrt{x+4}}\).

Solution

PatrickJMT - 439 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)=\sqrt{x-8}; g(x)=x^2}\).

Problem Statement

Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)=\sqrt{x-8}; g(x)=x^2}\).

Solution

PatrickJMT - 440 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)=\frac{1}{x}; }\) \(\displaystyle{ g(x)=\frac{1}{(x+2)(x-3)}}\).

Problem Statement

Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)=\frac{1}{x}; }\) \(\displaystyle{ g(x)=\frac{1}{(x+2)(x-3)}}\).

Solution

PatrickJMT - 438 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Really UNDERSTAND Precalculus

Log in to rate this page and to see it's current rating.

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

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.

more calculus help

Find the perfect fit with Prime Try Before You Buy

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon

Getting Started

Practice

Domain of Composite Functions

Practice

Practice Search

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 © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics