17Calculus

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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