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When math textbooks provide exercises in function composition, the typical exercise works as follows: the two functions are given, and the composition must be found. For example:

If f(x) = 3x + 1 and g(x) = x² - 1, what is f(g(x))?
 

Students then plug (x² - 1) into f: f(g(x)) = f(x² - 1) = 3(x² + 1) - 1 = 3x² + 3 - 1 = 3x² + 2.

What's often missing is the type of problem in which the composition is given, and the student must find two functions whose composition is the given function. This sort of problem is helpful when students reach calculus, and have to use the Chain Rule. The Chain Rule involves finding the derivate of a composite function, but we are rarely given the function in a composite form; we must identify two functions that form the composite function in order to use the Chain Rule.

Before expecting students to accomplish this task, we give them at least a couple sample problems with solutions.

Example #1

Find an f(x) and g(x) such that f(g(x)) =

x² - 1

If g(x) = x² - 1, then 

f(g(x)) =

So g(x) = x² - 1; f(x) =

Example #2

Find an f(x) and g(x) such that f(g(x)) = x + 7

x

+ 12

If g(x) = then f(g(x)) = g(x)² + 7g(x) + 12

Therefore, g(x) = ; f(x) = x² + 7x + 12

Example #3

Find an f(x) and g(x) such that f(g(x)) = x² + 8x + 16 +

1

2

(2x + 8)

f(g(x)) = (x + 4)² + (x + 4)

Noticing that we have a repeated (x + 4), this gives us the idea of setting g(x) = x + 4

Now the equation is: f(g(x)) = g(x)² + g(x)

Therefore, g(x) = x + 4; f(x) = x² + x

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This worksheet requires students to work backwards to find composite functions. 

NOTES

• These should be graded carefully, as there may be multiple answers to each question. I would recommend adding alternate answers to your own answer key, so you'll have them for future years.
• Students should be discouraged from using "trivial" functions. For example, setting g(x) = x makes each problem trivial

Lesson by Mr. H

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x² - x

x

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