# How to Decompose Functions ••• waveform 20 image by chrisharvey from Fotolia.com

Not all algebraic functions can simply be solved via linear or quadratic equations. Decomposition is a process by which you can break down one complex function into multiple smaller functions. By doing this, you can solve for functions in shorter, easier-to-understand pieces.

## Decomposing Functions

You can decompose a function of x, expressed as f(x), if a part of the equation can also be expressed as a function of x. For example:

f(x) = 1/(x^2 -2)

You can express x^2 - 2 as a function of x, and place this in f(x). You can call this new function g(x).

g(x) = x^2 - 2 f(x) = 1/g(x)

You can set f(x) as equal to 1/g(x) because the output of g(x) will always be x^2 - 2. But you can decompose this function further, by expressing 1 divided by a variable as a function. Call this function h(x):

h(x) = 1/x

You can then express f(x) as the two decomposed functions nested:

f(x) = h(g(x))

This is true because:

h(g(x)) = h(x^2 - 2) = 1/(x^2 - 2)

## Solving Using Decomposed Functions

Decomposed functions are solved from the inside out. Using f(x) = h(g(x)), you first solve for the g function, then the h function with the output of the g function.

For example, x = 4. First solve for g(4).

g(4) = 4^2 - 2 = 16 - 2 = 14

You then solve h using g's output, in this case, 14.

h(14) = 1/14

Since f(4) equals h(g(4)), f(4) equals 14.

## Alternate Decompositions

Most functions that can be decomposed can be decomposed in multiple ways. For instance, you could decompose f(x) using the following functions instead.

j(x) = x^2 k(x) = 1/(x - 2)

Placing j(x) as the variable for k(x) produces 1/(x^2 - 2), so:

f(x) = k(j(x))

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