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Imagine a factory assembly line: one machine shapes raw clay into a bowl, and the next machine paints it. If you feed clay into the combined system, it doesn't get painted until after it's shaped.
In algebra, a composite function works the exact same way by feeding the entire output of one function into another function as its input. We write this as (f∘g)(x)(f \circ g)(x) or f(g(x))f(g(x)), which we read aloud as "ff of gg of xx."
How do we build a single formula for this combined machine when both functions are algebraic equations?

The Substitution Strategy

To find f(g(x))f(g(x)), start by writing the outer function f(x)f(x) with empty parentheses wherever you see the variable xx. Then, drop the entire formula of the inner function g(x)g(x) right into every set of parentheses.
Let's walk through the algebra for f(x)=x2βˆ’3xf(x) = x^2 - 3x and g(x)=2x+1g(x) = 2x + 1. Expanding the squared binomial gives (2x+1)2=4x2+4x+1(2x + 1)^2 = 4x^2 + 4x + 1.
Next, distribute the βˆ’3-3 across (2x+1)(2x + 1) to get βˆ’6xβˆ’3-6x - 3. Combining like terms gives the simplified composite formula: f(g(x))=4x2βˆ’2xβˆ’2f(g(x)) = 4x^2 - 2x - 2.
What happens if we swap the machines and calculate g(f(x))g(f(x)) instead?

Order Changes the Outcome

Function composition is non-commutative, which means changing the order of composition almost always produces a completely different result: (f∘g)(x)β‰ (g∘f)(x)(f \circ g)(x) \neq (g \circ f)(x).
For g(f(x))g(f(x)), gg is now the outer function. Substituting f(x)=x2βˆ’3xf(x) = x^2 - 3x into g(x)=2x+1g(x) = 2x + 1 yields 2(x2βˆ’3x)+12(x^2 - 3x) + 1, which simplifies to 2x2βˆ’6x+12x^2 - 6x + 1.