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Function Composition: Decomposing a Composite Function

Calculus > Function Composition

function compositionIn order to figure out function composition (or to decompose a function), you must be familiar with the eight common function types and with basic function transformations, like:

  • A negative sign flips an axis around the origin),
  • Adding a constant shifts the function’s graph to the left that number of units.

Splitting a function into two can be useful if the original composite function is too complicated to work with. Composite functions are usually represented by f(x) and g(x), where f(x) is a function that takes some kind of action on g(x). For example:
f(g(x)) = -(x – 3)2 + 5
is a composite function with f(x) taking an action on g(x). The question becomes what function is f(x) and what function is g(x)?


Basic Function Types

Function decomposition is, in a very basic sense, splitting a complicated function into basic pieces. Those “basic pieces” are going to have one of the following eight forms. In order to be able to decompose a function, you must be able to recognize these forms.

Function Type Format Which terms are constants?
Exponential y = a bx a, b
Linear y = m x + b m, b
Logarithmic y = a ln (x) + b, a, b
Polynomial y = an · xn + an−1 · xn −1 + … + a2 · x2 + a1 · x + a0, an, an−1, … , a2, a1, a0
Power y = a xb a, b
Rational Ratio of two polynomials Same as polynomials
Quadratic y = a x2 + b x + c a,b,c
Sinusoidal y = a sin (b x + c), a,b,c

Function Composition Example Problem

Example problem 1: Identify the functions in the equation f(g(x)) = -(x – 3)2 + 5

Step 1: Identify the original function(s). The original function is either going to be: linear, polynomial, square (quadratic), absolute value, square root, rational, sine or cosine. In this example, the original function is the square – (x – 3)2 (the square – (x – 3) has been shifted up five units).

Step 2: Write the functions using standard terminology (f(x) and g(x)).
f(g(x)) = -(x – 3)2 + 5, so:
g(x) = – (x – 3)2
f(x) = x + 5

Example problem 2: Identify the functions in the equation f(g(x)) = (x + 2 / x)3

Step 1: Look for the original function f(x)—see Step 1 of example problem 1 above. In this example, the original function isn’t an obvious example of a basic function type. However, while the function f(x) = x + 2/x isn’t a basic type, the second function g(x)—x3 is (it’s a cubic polynomial). So:
f(g(x)) = (x + 2 / x)2
f(x) = x + 2 /x
g(x) = x3

Tip: When trying to find composite functions, look for the simplest transformation, usually involving x and a cube, square, simple addition, division, multiplication, subtraction etc.. This simple transformation is either going to be f(x) or g(x).

References

5.2 – Reference – Graphs of eight basic types of functions.

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Function Composition: Decomposing a Composite Function was last modified: June 14th, 2018 by Stephanie