Practice Composition Chains in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick 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.
Example 1
easyLet f(x)=x+1, g(x)=2x, h(x)=x^2. Compute (f\circ g\circ h)(3) step by step.
Example 2
mediumFind 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.
Example 3
easyGiven f(x)=3x-1 and g(x)=x^2, find (f\circ g)(x) and evaluate it at x=2.
Example 4
hardDecompose H(x)=\sin(e^{x^2}) as a composition H=f\circ g\circ h of three simpler functions.