2) Decomposition - Separating composite functions into smaller functions Assign one function to a where would you put parentheses? “cluster” – Ex c Express the following functions as the composition of functions. 1) h(x) = (2x + 3)3 2) h(x) = 3) h(x) =+ 4 (modified) possible to decompose in more than one way It’s Inverses and One-to-One Functions Function Never 2 outputs for same input Passes vertical line test no vertical line cuts graph twice – One-to-One does not apply if not a function – it’s Never has 2 inputs produce same output Passes horizontal line test no horizontal line cuts graph twice. – Which are functions? Which are one-to-one?: One-to-one functions have the properties Each f(x) has only one x associated with it, and each x has only one f(x) If f(x1) = f(x2), then x1 = x2 (if 2 outputs are the same, the inputs must be same) They are invertible (they have an inverse) , where f-1(x) is the inverse of f(x) Finding formulas for the inverse of a function We use the notation f-1(x) = inverse of or of (NOT an “the f(x)” “f-inverse x” exponent) Procedure for finding f-1(x) Let y = f(x) Exchange x and y (this creates the inverse relationship) Isolate y Replace y with f-1(x)
2) Decomposition - Separating composite functions into smaller functions Assign one function to a “cluster” – where would you put parentheses? Ex c Express the following functions as the composition of functions. 1) h(x) = (2x + 3)3 2) h(x) = 3) h(x) =+ 4 (modified) It’s possible to decompose in more than one way Inverses and One-to-One Functions Function Never 2 outputs for same input Passes vertical line test – no vertical line cuts graph twice One-to-One – does not apply if it’s not a function Never has 2 inputs produce same output Passes horizontal line test – no horizontal line cuts graph twice. Which are functions? Which are one-to-one?: One-to-one functions have the properties Each f(x) has only one x associated with it, and each x has only one f(x) If f(x1) = f(x2), then x1 = x2 (if 2 outputs are the same, the inputs must be same) They are invertible (they have an inverse) , where f-1(x) is the inverse of f(x) Finding formulas for the inverse of a function We use the notation f-1(x) = “the inverse of f(x)” or “f-inverse of x” (NOT an exponent) Procedure for finding f-1(x) Let y = f(x) Exchange x and y (this creates the inverse relationship) Isolate y Replace y with f-1(x)