Composition Chains Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Decompose H(x)=sin⁡(ex2)H(x)=\sin(e^{x^2}) as a composition H=f∘g∘hH=f\circ g\circ h of three simpler functions.

Solution

  1. 1
    Working from innermost to outermost: first h(x)=x2h(x)=x^2 (squaring), then g(x)=exg(x)=e^x (exponential), then f(x)=sin⁡(x)f(x)=\sin(x) (sine).
  2. 2
    Verify: (f∘g∘h)(x)=f(g(h(x)))=f(g(x2))=f(ex2)=sin⁡(ex2)=H(x)(f\circ g\circ h)(x)=f(g(h(x)))=f(g(x^2))=f(e^{x^2})=\sin(e^{x^2})=H(x). ✓

Answer

h(x)=x2h(x)=x^2, g(x)=exg(x)=e^x, f(x)=sin⁡(x)f(x)=\sin(x); H=f∘g∘hH=f\circ g\circ h
Decomposing a complex function into simpler components is crucial for applying the chain rule in calculus. The key is to identify successive 'layers' working from outside in.

About Composition Chains

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

Learn more about Composition Chains →

More Composition Chains Examples

Example 1 easy

Let [formula], [formula], [formula]. Compute [formula] step by step.

Example 2 medium

Find the formula for [formula] and [formula] where [formula] and [formula]. Show they are not equal.

Example 3 easy

Given [formula] and [formula], find [formula] and evaluate it at [formula].

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