Functions and Inverses What's the use? If f exists, we could use it to ...

Functions and InversesDefinition: A function g is said to be the inverse of f if:1. domain of f = range of g,2. range of f = domain of g, and3. y = f(x) if and only if g(y) = x.Notation: f ­1 is the inverse of f.What's the use? If f ­1 exists, we could use it to solveequations like f(x) = 0, because then...Jan 28­1:47 PMFunctions and InversesVisual Calculus ­ Inverses of Functions (Overview)Visual Calculus ­ Inverses of Functions (Flash Tutorial)Jan 28­2:01 PM1

Examples ­ Decide whether f is one­to­one.­ either use horizontal line test­ or apply definition­ be sure to specify the domain!Jan 28­4:37 PMGiven a graph of y = f(x), we can sketch the inverse:f ­1 (x) = (x 2 ­2)/5a) Sketch f ­1 (x).b) State the domain and range.c) Find a formula for f ­1 (x).Jan 28­4:59 PM2

Find the inverse, f ­1 (x).Jan 28­4:45 PMInverse Trig FunctionsDoes f(x) = sin(x) have an inverse?Does cos(x)? Does tan(x)?Jan 29­1:02 PM3

Important Calc II Skill:(used w/ integration method of trig. subst.)(a) Evaluate cos(sin ­1 (x))(b) Evaluate tan(cos ­1 (x))Jan 29­1:06 PMDerivatives of Inverse Trig Functions(relies on implicit differentiation)Formula: d/dx [sin ­1 (x)] = 1/sqrt[1­x 2 ]Proof:Caution: the derivative,not sin ­1 itself!Jan 29­1:22 PM4