Step By Step Calculus » 4.5 - Composite Functions

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Synopsis
A composite function is created by the successive application (= composition) of one or more functions. For example, h(x)=\sqrt{|x|-1}h(x)=\sqrt{|x|-1} satisfies h(x)=f(g(x))h(x)=f(g(x)), where f(u)=\sqrt{u}f(u)=\sqrt{u} and g(x)=\vert x \vert -1g(x)=\vert x \vert -1. In other words, the map hh is the application of the two functions g(x) = |x| - 1g(x) = |x| - 1 (first) and f(u) = \sqrt{u}f(u) = \sqrt{u} (second). Notation: f(g(x)) f(g(x)) is often written f \circ g (x) f \circ g (x) .
When a function hh is defined as h(x)=f\circ g(x)h(x)=f\circ g(x), its properties can be derived from the properties of the two functions ff and gg. If the domain of hh is D_hD_h, and the domains of ff and gg are D_fD_f and D_gD_g respectively while the ranges are E_{h}, \; E_{f}, E_{g} E_{h}, \; E_{f}, E_{g} respectively, then
  • D_{h}=\left\{ x\in D_{g} \; | \; g(x)\in D_{f}\right\} D_{h}=\left\{ x\in D_{g} \; | \; g(x)\in D_{f}\right\} .
  • E_{h}=\left\{ f(g(x)) \; | \; x\in D_{g}, \, g(x) \in D_{f} \right\} E_{h}=\left\{ f(g(x)) \; | \; x\in D_{g}, \, g(x) \in D_{f} \right\} , or equivalently E_{h}=\left\{ f(y) \; | \; y\in E_{g} \cap D_{f} \right\} E_{h}=\left\{ f(y) \; | \; y\in E_{g} \cap D_{f} \right\} .