Composition of Functions

You already know how to add, subtract, and multiply functions. But there’s another way to combine them that’s even more powerful: composition. Instead of combining outputs, you feed the output of one function directly into another. It’s like an assembly line — the first machine processes the input, then hands its result to the second machine.

1. The Basic Idea: f(g(x))

When we write f(g(x)), we mean: first evaluate g at x, then plug that result into f. The “inner” function g goes first, and the “outer” function f goes second.

Let’s start with two simple functions and see what their composition looks like.

The purple line is f, the blue curve is g, and the red curve is f(g(x)). For any x-value, the red curve’s height equals f evaluated at whatever g(x) is. For example, at x = 2: g(2) = 4, then f(4) = 2(4) + 1 = 9.

2. Adjustable Composition

Now let’s make it interactive. Control the parameters of both functions and watch the composition update in real time.

f(x) = a*x + b and g(x) = c*x^2 + d

3. Order Matters: f(g(x)) vs g(f(x))

This is the most important thing about composition: it is NOT commutative. Swapping the order usually gives a completely different function. Let’s see the proof.

4. Composing with Trig and Polynomial

Composition isn’t limited to polynomials. Let’s compose a trig function with a linear one. Here f(x) = sin(x) and g(x) = mx + c. The composition f(g(x)) = sin(mx + c) is exactly the general sine wave from trigonometry!

5. Decomposing a Complex Function

Going backwards is just as important: given a complicated function, can you break it into simpler pieces? For example, h(x) = (3x - 1)^2 can be seen as f(g(x)) where g(x) = 3x - 1 and f(x) = x^2.