AI Tutor
If you build a complex Lego set, you assemble small bricks into larger sub-assemblies. Function decomposition is simply running that assembly line in reverse to see which basic pieces created the final shape.
Instead of combining two functions ff and gg to create h(x)=f(g(x))h(x) = f(g(x)), our goal is to start with h(x)h(x) and break it apart into an inner function and an outer function.
How do we know which piece is on the inside and which piece is on the outside?

The Inside-Out Architecture

Think of h(x)=f(g(x))h(x) = f(g(x)) as a nested container. The inner function, g(x)g(x), is the operation applied first directly to the input xx, while the outer function, f(u)f(u), acts on the entire output of g(x)g(x).
What visual clues in an algebraic formula tell us where the inner function ends and the outer function begins?

Spotting the Inner Function

Look for expressions trapped inside common algebraic 'containers': parentheses raised to a power, expressions under a radical sign, expressions in a denominator, or arguments of trigonometric functions.
Let's decompose h(x)=5x3−7h(x) = \sqrt{5x^3 - 7}. The expression tucked under the square root radical is 5x3−75x^3 - 7.
Setting g(x)=5x3−7g(x) = 5x^3 - 7 and f(x)=xf(x) = \sqrt{x}, we verify by evaluating f(g(x))=f(5x3−7)=5x3−7f(g(x)) = f(5x^3 - 7) = \sqrt{5x^3 - 7}, which matches h(x)h(x) exactly.
Can a function have more than one valid pair of components?