Function Inverses

Function Inverses Lecture: 2EPre AP & GT Precalculus

Agenda • 12 minute, 11 Question Quiz • One to One Functions • Inverse Functions

Injective Functions • (definition) A function for which each member of its range is associated with one and only one member of its domain. • Commonly called One-to-One Functions

One-to-One

Numerical Examples One-to-One? NO YES NO

Horizontal Line Test One-to-One Not One-to-One

Reflections Sometimes, Always, Never • An even function is _____ one-to-one • Never: Even functions don’t pass Hor. Line Test • An odd function is ______ one-to-one • Always: Odd Functions pass Hor. Line Test

The Inverse Function • If then • Hint: • Answer: _

Possible Notation Confusion • In the notation , -1 is not an exponent. • Inverse is not the same as reciprocal. • Example: • Inverse: Reciprocal:

Graphical Inverse • A function and its inverse are symmetric about the line y=x

Example: Are they inverses? NO YES

Reflection • If (x,y) is on the graph of f(x), then which point is on the graph of the inverse of f(x)? • (y,x)

Finding Inverse: Algebraic Procedure Example 1. 2. 3. • 1. Start with function • 2. Switch y and x • 3. Solve for y

Algebraic Examples

Reflection • Does for all functions f? • No, counterexample when • Does for all one-to-one functions? • YES, this is known as the inverse composition rule

Connections Inverse Composition Rule • To show that two functions, f and g, are inverses algebraically: • Show: AND • Show:

Homework • Pg 99 V1-5, • Pg 99 #9-12(A), 24 • Pg 100 #25-32(A) • Pg 100 #60, 64 • Pg102 #85,86,93,94 • Test next Tuesday

Function Inverses