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 what a negative sign does to a function (flips an axis around the origin) or what happens if you add a constant (it shifts to the left that number of units). In calculus, 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). Being able to split a function into two can be useful in calculus if the original function is too complicated to work with.

## Function Composition Sample Problem

**Sample 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

**Sample 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 sample 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).

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