4.1 Inverses Mon March 23 Do Now Solve 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 Compositions Tues 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 of and show they are inverses using compositions HW: p.358 #67-87 odds