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 () = 2 − 3 and () = 2 − 1: Sum of functions: ( + )() = () + () o () + () = (2 − 3) + ( 2 − 1) = 2 + 2 − 4 o Add the two functions together and combine like terms Difference of functions: ( − )() = () − () o () − () = (2 − 3) − ( 2 − 1) = − 2 + 2 − 2 o Subtract the two functions, REMEMBER TO PUT THE SECOND IN PARENTHESIS AND DISTRIBUTE THE NEGATIVE, then combine like terms Product: ()() = () ∙ () o ()() = (2 − 3)( 2 − 1) = 2 3 − 3 2 − 2 + 3 o FOIL the two functions together Division:( ) () = o () () = 2−3 2 −1 () () , ≠ ±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. () = + & () = √ − B. () = √ − & () = a.) ( + )() a.) ( + )() b.) ( − )() b.) ( − )() c.) ()() c.) ()() d. ( ) () and find the domain FROM A GRAPH: d. ( ) () and find the domain + Composition of Functions The composition of the function f with the function g is: ( ∘ )() = (()) Which is read f of g of x. The domain of ( ∘ ) is the set of all x in the domain of g such that g(x) is the domain of f. Given () = + 2 & () = 4 − 2 : a.) ( ∘ )() b.) ( ∘ )() c.) ( ∘ )() d.) ( ∘ )(−2)

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# 2.6 – Notes on Combinations of Functions