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.

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).

Okay, time for some practice problems.

* 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

| |

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 |

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.