Combinations of Functions & Inverse Functions Obj: Be able to work with combinations/compositions of functions. Be able to find inverse functions. TS: Making decisions after reflections and review Warm-Up: Given f(x) = 2x + 3 and g(x) = x2 – 4, find… f(2) + g(2) g(3) – f(3)

Arithmetic Combinations Where the domain is the real numbers that both f and g’s domains have in common. For f/g also g(x) ≠ 0.

Examples Find each of the below combinations given (f∙g)(2) 3) (f/g)(x) & its domain (f – g)(2)

Arithmetic Compositions (f○g)(x) = f(g(x)) Given the below find each of the following. (f ○ g)(2) 2) g(f(-1))

Arithmetic Compositions (f○g)(x) = f(g(x)) Given the below find each of the following. 3) f(g(x)) & its domain

Find f(g(x)) and g(f(x))

Definition of the Inverse of a Function Let f and g be two functions where f(g(x)) = x for every x in the domain of g, and g(f(x)) = x for every x in the domain of f. Under these conditions, g is the inverse of f and g is denoted f-1. Thus f(f-1(x))=x and f-1(f(x))=x where the domain of f must equal the range of f-1 and the range of f must equal the domain of f-1.

Graphs of Inverses Two equations are inverses if their graphs are reflections of one another across the line y=x. y f y=x f -1 1 x 1

Inverse FUNCTIONS A function f(x) has an inverse function if the graph of f(x) passes the ___________________. (In other words the relation is ONE-TO-ONE: For each y there is exactly one x) Circle the functions that are one-to-one (aka have inverse functions) y y y y x x x x y x

Examples: Determine if the two functions f and g are inverses. 1) and

Finding an Inverse Verify that the function is one-to-one thus has an inverse function using the Horizontal Line Test. Switch x & y. Solve for y. Make sure to use proper inverse notation for y for your final answer. (Ex: f-1(x), not y) To check your answer: Verify they are inverses by testing to see if f(f-1(x)) = f-1(f(x)) = x

Find the inverse function if there is one, if there is not one, restrict the domain to make it one-to-one then find the inverse function . f(x) = -.5x + 3

Find the inverse function if there is one, if there is not one, restrict the domain to make it one-to-one then find the inverse function . 2) f(x) = x2 – 4