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 this 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.
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.
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).
Time for some practice.
Unless otherwise instructed, for \( f(x) = 3x  1 \) and \( g(x) = 2x^2 + x + 1 \), find \( f \circ g \) and \( g \circ f \).
Problem Statement 

Unless otherwise instructed, for \( f(x) = 3x  1 \) and \( g(x) = 2x^2 + x + 1 \), find \( f \circ g \) and \( g \circ f \).
Solution 

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Unless otherwise instructed, for \( f(x) = 3x  4 \) and \( g(x) = x^3  3 \), find \( f \circ g \) and \( g \circ f \).
Problem Statement 

Unless otherwise instructed, for \( f(x) = 3x  4 \) and \( g(x) = x^3  3 \), find \( f \circ g \) and \( g \circ f \).
Solution 

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Unless otherwise instructed, for \( f(x) = 5x + 2 \) and \( g(x) = x^3  4 \), calculate \( f(g(2)) \) and \( g(f(1)) \).
Problem Statement 

Unless otherwise instructed, for \( f(x) = 5x + 2 \) and \( g(x) = x^3  4 \), calculate \( f(g(2)) \) and \( g(f(1)) \).
Solution 

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Unless otherwise instructed, for \( f(x) = 2x  3 \) and \( g(x) = 5x + 1 \), find \( f \circ g \).
Problem Statement 

Unless otherwise instructed, for \( f(x) = 2x  3 \) and \( g(x) = 5x + 1 \), find \( f \circ g \).
Solution 

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Unless otherwise instructed, for \( f(x) = x + 3 \) and \( g(x) = x^2  5 \), find \( f \circ g \) and \( g \circ f \).
Problem Statement 

Unless otherwise instructed, for \( f(x) = x + 3 \) and \( g(x) = x^2  5 \), find \( f \circ g \) and \( g \circ f \).
Solution 

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Unless otherwise instructed, for \( f(x) = 3x + 2 \) and \( g(x) = x^2 + 1 \), find \( f \circ g \) and \( g \circ f \).
Problem Statement 

Unless otherwise instructed, for \( f(x) = 3x + 2 \) and \( g(x) = x^2 + 1 \), find \( f \circ g \) and \( g \circ f \).
Solution 

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Unless otherwise instructed, for \( f(x) = x^2  3 \) and \( g(x) = \sqrt{x1} \), find \( f \circ g \) and \( g \circ f \).
Problem Statement 

Unless otherwise instructed, for \( f(x) = x^2  3 \) and \( g(x) = \sqrt{x1} \), find \( f \circ g \) and \( g \circ f \).
Solution 

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

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

video by PatrickJMT 

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For \(f(x)=x^22x+3\), \(g(x)=2x+1\), find \(f \circ g\) and \(g \circ f\).
Problem Statement 

For \(f(x)=x^22x+3\), \(g(x)=2x+1\), find \(f \circ g\) and \(g \circ f\).
Solution 

video by MIP4U 

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\(f(x)=2x3\), \(g(x)=6x^24x+5\), find \(f(g(7))\), \(g(f(2))\), \(f(g(3))\), \(f(f(4))\).
Problem Statement 

\(f(x)=2x3\), \(g(x)=6x^24x+5\), find \(f(g(7))\), \(g(f(2))\), \(f(g(3))\), \(f(f(4))\).
Solution 

video by MIP4U 

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For \(f(x)=x^2+5\), \(g(x)=6x15\), find \((f \circ g)(2)\) and \((g \circ f)(2)\).
Problem Statement 

For \(f(x)=x^2+5\), \(g(x)=6x15\), find \((f \circ g)(2)\) and \((g \circ f)(2)\).
Solution 

video by MIP4U 

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For \(f(x)=3x/5+4\), \(g(x)=2x^25x+9\), find \((f \circ g)(2)\), \((g \circ f)(4)\), \((f \circ g)(1/2)\), \((f \circ f)(4/5)\).
Problem Statement 

For \(f(x)=3x/5+4\), \(g(x)=2x^25x+9\), find \((f \circ g)(2)\), \((g \circ f)(4)\), \((f \circ g)(1/2)\), \((f \circ f)(4/5)\).
Solution 

video by MIP4U 

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For \(f(x)=1/(x+5)\), \(g(x)=3/x5\), find \(f \circ g\) and \(g \circ f\).
Problem Statement 

For \(f(x)=1/(x+5)\), \(g(x)=3/x5\), find \(f \circ g\) and \(g \circ f\).
Solution 

video by MIP4U 

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

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

video by Krista King Math 

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For \(h(x)=3x\), \(g(t)=2t2h(t)\), \(f(n)=5n^2+h(n)\), find \(h(g(8))\).
Problem Statement 

For \(h(x)=3x\), \(g(t)=2t2h(t)\), \(f(n)=5n^2+h(n)\), find \(h(g(8))\).
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.
Unless otherwise instructed, 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 

Unless otherwise instructed, 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 

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Unless otherwise instructed, for \( f(x) = \sqrt{x2} \) and \( g(x) = 1/(x+3) \), find \( f \circ g \) and it's domain.
Problem Statement 

Unless otherwise instructed, for \( f(x) = \sqrt{x2} \) and \( g(x) = 1/(x+3) \), find \( f \circ g \) and it's domain.
Solution 

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Unless otherwise instructed, for \( f(x) = \sqrt{x5} \) and \( g(x) = 1/(8x) \), find \( g \circ f \) and it's domain.
Problem Statement 

Unless otherwise instructed, for \( f(x) = \sqrt{x5} \) and \( g(x) = 1/(8x) \), find \( g \circ f \) and it's domain.
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 

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 

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 

video by PatrickJMT 

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Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)=\sqrt{x8}; g(x)=x^2}\).
Problem Statement 

Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)=\sqrt{x8}; g(x)=x^2}\).
Solution 

video by PatrickJMT 

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Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)=\frac{1}{x}; }\) \(\displaystyle{ g(x)=\frac{1}{(x+2)(x3)}}\).
Problem Statement 

Find the domain of \(g(f(x))\) where \(\displaystyle{f(x)=\frac{1}{x}; }\) \(\displaystyle{ g(x)=\frac{1}{(x+2)(x3)}}\).
Solution 

video by PatrickJMT 

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