Math 1304 Calculus I 1.6 Inverse Functions

1.6 Inverse functions Definition: A function f is said to be one-to- one if f(x) = f(y) implies x = y. It never takes on the same value twice Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once.

Inverse Functions Definition: Let f and g be functions. They are said to be inverse if y = f(x) ↔ g(y) = x Theorem: If f is a one-to-one function then it has an unique inverse. Notation: the inverse of f is denoted by f -1

Rules for inverses f -1 (f(x)) = x, for all x in the domain of f f (f -1 (x)) = x, for all x in the domain of f -1

Finding an inverse Write y = f(x) and solve for x in terms of y.

Logarithms are inverse to exponentials log a (y) = x iff y = a x

Laws for logarithms See page 64

Natural Logarithms Natural = base e ln(x) = log e (x)

The number e e = … is a special number that is used as a base for exponential functions in calculus