Composition Chains Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Composition Chains.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A composition chain is a sequence of functions applied one after another: (f \circ g \circ h)(x) = f(g(h(x))), evaluated inside-out from right to left.

Work from the innermost function outward — compute h(x) first, then feed that result to g, then feed that to f. The order matters critically.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Composition is not commutative: f(g(x)) \neq g(f(x)) in general.

Common stuck point: Apply functions inside-out: g first, then f, in f(g(x)).

Sense of Study hint: Write out each intermediate result on a separate line: first h(x) = ?, then g(that) = ?, then f(that) = ?. Don't skip steps.

Worked Examples

Example 1

easy
Let f(x)=x+1, g(x)=2x, h(x)=x^2. Compute (f\circ g\circ h)(3) step by step.

Solution

  1. 1
    Start from the innermost function: h(3)=3^2=9.
  2. 2
    Apply g: g(h(3))=g(9)=2(9)=18.
  3. 3
    Apply f: f(g(h(3)))=f(18)=18+1=19. So (f\circ g\circ h)(3)=19.

Answer

(f\circ g\circ h)(3) = 19
Function composition chains are evaluated from right to left (innermost to outermost). The output of each function becomes the input of the next. Keeping track of this order is essential for correct computation.

Example 2

medium
Find the formula for (g\circ f)(x) and (f\circ g)(x) where f(x)=x^2+1 and g(x)=\sqrt{x}. Show they are not equal.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Given f(x)=3x-1 and g(x)=x^2, find (f\circ g)(x) and evaluate it at x=2.

Example 2

hard
Decompose H(x)=\sin(e^{x^2}) as a composition H=f\circ g\circ h of three simpler functions.
← Back to Composition Chains Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

composition