advertisement

2.6 – Combinations of Functions We can add, subtract, multiply and divide functions. We can find the composition of one function with another and use them to model and solve real-life problems. Arithmetic Combinations Using 𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) = 𝑥 2 − 1: Sum of functions: (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥) o 𝑓(𝑥) + 𝑔(𝑥) = (2𝑥 − 3) + (𝑥 2 − 1) = 𝑥 2 + 2𝑥 − 4 o Add the two functions together and combine like terms Difference of functions: (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥) o 𝑓(𝑥) − 𝑔(𝑥) = (2𝑥 − 3) − (𝑥 2 − 1) = −𝑥 2 + 2𝑥 − 2 o Subtract the two functions, REMEMBER TO PUT THE SECOND IN PARENTHESIS AND DISTRIBUTE THE NEGATIVE, then combine like terms Product: (𝑓𝑔)(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥) o (𝑓𝑔)(𝑥) = (2𝑥 − 3)(𝑥 2 − 1) = 2𝑥 3 − 3𝑥 2 − 2𝑥 + 3 o FOIL the two functions together 𝑓 Division:( ) (𝑥) = o 𝑔 𝑓(𝑥) 𝑔(𝑥) = 2𝑥−3 𝑥 2 −1 𝑓(𝑥) 𝑔(𝑥) , 𝑥 ≠ ±1 o Write the ratio of the two functions as a fraction o Remember to inspect the domain of each to find the excluded x-values A. 𝒇(𝒙) = 𝒙𝟐 + 𝟔 & 𝒈(𝒙) = √𝟏 − 𝒙 B. 𝒇(𝒙) = √𝒙𝟐 − 𝟒 & 𝒈(𝒙) = a.) (𝑓 + 𝑔)(𝑥) a.) (𝑓 + 𝑔)(𝑥) b.) (𝑓 − 𝑔)(𝑥) b.) (𝑓 − 𝑔)(𝑥) c.) (𝑓𝑔)(𝑥) c.) (𝑓𝑔)(𝑥) 𝑓 d. ( ) (𝑥) and find the domain 𝑔 FROM A GRAPH: 𝑓 d. ( ) (𝑥) and find the domain 𝑔 𝒙𝟐 𝒙𝟐 +𝟏 Composition of Functions The composition of the function f with the function g is: (𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥)) Which is read f of g of x. The domain of (𝑓 ∘ 𝑔) is the set of all x in the domain of g such that g(x) is the domain of f. Given 𝑓(𝑥) = 𝑥 + 2 & 𝑔(𝑥) = 4 − 𝑥 2 : a.) (𝑓 ∘ 𝑔)(𝑥) b.) (𝑔 ∘ 𝑓)(𝑥) c.) (𝑓 ∘ 𝑓)(𝑥) d.) (𝑔 ∘ 𝑓)(−2)