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).

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**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))**