Music Composition For Dummies
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A composition of functions is one function acting upon another. Think of it like putting one function inside of the other — f(g(x)), for instance, means that you plug the entire g(x) function in for all x’s in f(x). To solve such a problem, you work from the inside out:

f(g(x)) = f(3x2 – 10) = (3x2 – 10)2 – 6(3x2 – 10) + 1

This process puts the g(x) function into the f(x) function everywhere the f(x) function asks for x. This equation ultimately simplifies to 9x4 – 78x2 + 161, in case you’re asked to simplify the composition (which you usually are).

Likewise,

How to break down a composition of functions.

which easily simplifies to 3(2x – 1) – 10 because the square root and square cancel each other. This equation simplifies even further to 6x – 13.

You may also be asked to find one value of a composed function. To find

Finding one value of a composed function

for instance, it helps to realize that it’s like reading Hebrew: You work from right to left. In this example, you’re asked to put –3 in for x in f(x), get an answer, and then plug that answer in for x in g(x). Here are these two steps in action:

f(–3) = (–3)2 – 6(–3) + 1 = 28

g(28) = 3(28)2 – 10 = 2,342

About This Article

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About the book authors:

Scott Jarrett is a producer and musician who currently runsthe Monkey House Recording Studio. Holly Day is the coauthor of Music Theory For Dummies and Music Composition For Dummies. Her articles have appeared in publications across the globe.

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