Calculus > Function Composition

In 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 b^{x} |
a, b |

Linear | y = m x + b | m, b |

Logarithmic | y = a ln (x) + b, | a, b |

Polynomial | y = a_{n} · x^{n} + a_{n−1} · x^{n −1} + … + a_{2} · x^{2} + a_{1} · x + a_{0}, |
a_{n}, a_{n−1}, … , a_{2}, a_{1}, a_{0} |

Power | y = a x^{b} |
a, b |

Rational | Ratio of two polynomials | Same as polynomials |

Quadratic | y = a x^{2} + 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)—x^{3} *is* (it’s a cubic polynomial). So:

f(g(x)) = (x + 2 / x)^{2}

f(x) = x + 2 /x

g(x) = x^{3}

**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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