Composite Functions Inverse Functions Section 6.1 Section 6.2 Composite Functions Inverse Functions

THE COMPOSITE FUNCTION Given two function f and g, the composite function, denote by f ◦ g (read “f composed with g”), is defined by ( f ◦ g)(x) = f (g(x)) The domain of f ◦ g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

CONCEPT OF AN INVERSE FUNCTION Idea: An inverse function takes the output of the “original” function and tells from what input it resulted. Note that this really says that the roles of x and y are reversed.

MATHEMATICAL DEFINITION OF INVERSE FUNCTIONS In the language of function notation, two functions f and g are inverses of each other if and only if

NOTATION FOR THE INVERSE FUNCTION We use the notation for the inverse of f(x). NOTE: does NOT mean

ONE-TO-ONE FUNCTIONS A function is one-to-one if for each y-value there is only one x‑value that can be paired with it; that is, each output comes from only one input.

ONE-TO-ONE FUNCTIONS AND INVERSE FUNCTIONS Theorem: A function has an inverse if and only if it is one-to-one.

TESTING FOR A ONE-TO-ONE FUNCTION Horizontal Line Test: A function is one-to-one (and has an inverse) if and only if no horizontal line touches its graph more than once.

GRAPHING AN INVERSE FUNCTION Given the graph of a one-to-one function, the graph of its inverse is obtained by switching x- and y-coordinates. The resulting graph is reflected about the line y = x.

FINDING A FORMULA FOR AN INVERSE FUNCTION To find a formula for the inverse given an equation for a one-to-one function: Replace f (x) with y. Interchange x and y. Solve the resulting equation for y. Replace y with f -1(x).