Precalculus · Problem 5.1.29

Further Topics in Functions — Stitz & Zeager

Mathematics — Precalculus · function-composition-triple

Problem

Use f(x)=−2xf(x) = -2x, g(x)=xg(x) = \sqrt{x} and h(x)=∣x∣h(x) = |x| to find and simplify an expression for (f∘h∘g)(x)(f \circ h \circ g)(x) and state the domain using interval notation.
Intuitive explanation

Function composition chains transformations from the inside out—here you'll apply gg first, then hh, then ff to build up the final rule. The key is working carefully through each layer: substitute the output of one function as the input to the next, then simplify. Don't forget that each function carries domain restrictions (square roots need non-negative inputs, absolute value is unrestricted), so your final domain is the intersection of all constraints that actually matter.

Intuitive explanation

Function composition chains transformations from the inside out—here you'll apply gg first, then hh, then ff to build up the final rule. The key is working carefully through each layer: substitute the output of one function as the input to the next, then simplify. Don't forget that each function carries domain restrictions (square roots need non-negative inputs, absolute value is unrestricted), so your final domain is the intersection of all constraints that actually matter.

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