In Precalculus composition comes with a domain to find and functions to take apart. The domain of f(g(x)) needs x to work in g first, and then g(x) to work in f. Writing a function as a composition of two simpler ones is the reverse skill.
Also taught in Algebra 2, at its own level and with its own examples.
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Pick more than one for a sheet that mixes them.
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Find the domain of f(g(x))for f(x)=x−21and g(x)=x
g needs x ≥ 0.
f(g(x)) = 1/(√x - 2), and f cannot take an input of 2.
√x = 2 when x = 4, so remove 4.
The answer is [0,4)∪(4,∞).
Where students go wrong
Finding the domain only from the simplified formula. Any x that the inside function rejects stays out of the domain, even if the final expression would accept it.
Also called composite functions, f of g of x or composing functions.
What you can put on this worksheet
Compose two functions
Given f(x)=x+3and g(x)=−3x−3, find f(g(x))→ −3x
Write h as f(g(x))
If h(x)=f(g(x))and h(x)=x+6, find f(x)and g(x), neither equal to x→ f(x)=x, g(x)=x+6
At a number
f(x)=3x+3, g(x)=−2x+1. Find g(f(−2))→ 7
A quadratic outside, or f(f(x))
f(x)=2x2+3and g(x)=−2x+1. Find f(g(x))→ 8x2−8x+5
From a table
Use the table to find g(f(2)). xf(x)g(x)132233331414→ 1
Questions about these worksheets
Yes. Every composition of functions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the composition of functions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 5: compose two functions, write h as f(g(x)), at a number, a quadratic outside, or f(f(x)) and from a table. Tick as many as you want and set how many of each, or let it spread them evenly.
Multiplying the two functions instead of nesting them. f(g(x)) is not f(x) times g(x). And swapping the order changes the answer, so read which one is inside.