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

• loga(y) = x iff y = ax

Laws for logarithms

• See page 64

Natural Logarithms

• Natural = base e• ln(x) = loge(x)

The number e

• e = 2.718281828… is a special number that is used as a base for exponential functions in calculus