 `2.3 The Composition of Functions p. 1 Composition of Functions If f and g are functions, then the composite function, or composition, of f and g is (𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥 )) for all x in the domain of f such that g(x) is in the domain of f. Evaluating Composite Functions Example Given 𝑓 (𝑥 ) = 2𝑥 − 1 and 𝑔(𝑥 ) = ./0, find (a) (𝑓 ∘ 𝑔)(2) (b) (𝑔 ∘ 𝑓)(−3) Finding Composite Functions Let 𝑓(𝑥 ) = √𝑥 and 𝑔(𝑥 ) = 2 − 3𝑥. Find: (a) (𝑔 ∘ 𝑓)(𝑥) Domain of (𝑔 ∘ 𝑓)(𝑥) (b) (𝑓 ∘ 𝑔)(𝑥) Domain of (𝑓 ∘ 𝑔)(𝑥) 2.3 The Composition of Functions p. 2 Finding Functions that Form a Composite Function Suppose that ℎ(𝑥 ) = (𝑥 4 − 5)6 − 4(𝑥 4 − 5) + 3. Find f and g such that (𝑓 ∘ 𝑔)(𝑥) = ℎ(𝑥) Application of Composition of Functions • Suppose an oil well off the California coast is leaking. – Leak spreads in circular layer over water – Area of the circle is 𝐴(𝑟) = 𝜋𝑟 4 • At any time t, in minutes, the radius increases 5 feet every minute. – Radius of the circular oil slick is 𝑟(𝑡) = 5𝑡 • Express the area as a function of time using substitution. `