How to Decompose Functions

Two functions
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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))