4.1 InversesMon March 23

Do NowSolve for Y

1)

2)

Quiz Review

• Retakes?

Inverses

• When we go from an output of a function back to its inputs, we get an inverse relation

• Interchanging the first and second coordinates of each ordered pair produces the inverse relation

Notation

• The notation for the inverse of f(x) is

• This is not an exponent!

One-to-one Functions

• A function f is one-to-one if different inputs have different outputs

• If

• Every y-value is unique

Properties of one-to-one functions

• If a function f is one-to-one, its inverse is a function

• The domain of a one-to-one function f is the range of its inverse

• The range of a one-to-one function is the domain of its inverse

• A function that is always increasing or decreasing is one-to-one

Ways to show a function is one-to-one

• 1) Assume f(a) = f(b); then show that a = b– If you can think of a y-value that has 2 x-values (ex:

x^2

• 2) Horizontal Line Test– If a horizontal line intersects the graph more than

once, it is NOT one-to-one

How to find a formula for inverse

• 1) Replace f(x) with y (if possible) • 2) Switch x and y• 3) Solve for y• 4) Replace y with

Ex

• Find an equation for the inverse of the relation

Ex2

• Find an inverse for the function

You try

• Find an inverse for the following• 1) f(x) = 7 - x

• 2)

• 3)

Closure

• Find the inverse for the function

• HW: p.356 #17-59 odds

4.1 Inverses and CompositionsTues March 24

• Do Now• Find the inverse of

HW Review: p.356 #17-59 odds

Inverses and Graphs

• The graph of an inverse function is a reflection of f(x)’s graph across the line y = x

Inverse Functions and Compositions

• If a function f(x) is one-to-one, then the following compositions are true:

for each x in the domain of f

for each x in the domain of f inverse

Ex

• Given f(x) = 5x + 8, find its inverse and show they are inverses using composition

Ex

• Given, find its inverse and show they are

inverses using composition

Restricting the domain

• In the case in which the inverse of a function is not a function, the domain can be restricted to allow the inverse to be a function

• This is why square root graphs only show half of the true graph in a calculator

Closure

• Find the inverse ofand show they are inverses using

compositions

HW: p.358 #67-87 odds