Warm Up Decide whether these are one to one functions, if so give the inverses. 𝑓 𝑥 = 15 11 𝑥+ 5 6 𝑦= 4 5𝑥 +2 𝑔 𝑥 = 𝑥−3 +1 𝑦= 𝑥 4 −5 Solutions: 𝑓 −1 𝑥 = 11 𝑥− 5 6 15 𝑓 −1 𝑥 = 4 5 𝑥−2 Not “one to one” – fails HLT

Graphing Inverses

Learning targets How to graph an inverse function Interpreting a data table of an inverse Finding an inverse of a non-invertible function using a graph

RECAP A function is a relationship between inputs and outputs where each input is related to only one output In order for a function to have an inverse it must be “one to one” This means that each input is related to one unique output only We can use the VLT and HLT to prove the above when looking at a functions graph.

REcap Inverses are functions that “undo” the original function We use the following notation to show that something is an inverse: 𝑓 −1 (−) To figure out a functions inverse we need to solve for x in terms of y

Graphs We will be looking at the following function and its inverse. 𝑓 𝑥 =2𝑥+1

Graphs Now we will find the inverse: 𝑓 𝑥 =2𝑥+1 𝑦=2𝑥+1 𝑦−1=2𝑥 𝑦−1 2 =𝑥 𝑓 −1 𝑦 = 𝑦−1 2

Lets just test it to see if it is the inverse: Graphs Lets just test it to see if it is the inverse: 𝑓 𝑥 =2𝑥+1 𝑓 1 =2 1 +1 𝑓 1 =3 𝑓 −1 𝑦 = 𝑦−1 2 𝑓 −1 3 = 3 −1 2 𝑓 −1 3 = 2 2 =1 IT WORKED!!!

Lets make a table What do you notice about the two tables? 𝑥 𝒇(𝑥) 1 3 1 3 2 5 7 4 9 11 𝑥 𝒇 −𝟏 (𝒙) -0.5 1 2 0.5 3 4 1.5 5 What do you notice about the two tables?

Graphs 𝑓 −1 𝑦 = 𝑦−1 2 is just a rewritten form of 𝑓 𝑥 =2𝑥+1 What did our inverse actually do? It swaps the x and y values. Switch x and y in the final inverse form… 𝑥= 𝑦−1 2 𝑦= 𝑥−1 2

Graphs Now lets look at the inverse functions graph: 𝑦= 𝑥−1 2

Graphs What do you notice about the graphs?

You Try Graph the function: 𝑓(𝑥)=0.2𝑥−2 Reflect it over the function 𝑦=𝑥 Is this the same graph as your inverse? Lets find out… Solve for your inverse function and graph it.

It should look like this…

Graph the Following Graph the following function and reflect it over the line y=x. Then solve for the inverse and plot it. 𝑓 𝑥 = (𝑥−1) 3

Graph “Undo” Function Remember… When graphing and determining inverse equations we can always check our answer to be sure… Graph “Undo” Function

Homework: Worksheet (do on a separate sheet of paper) You should be able to: Determine if a function has an inverse Know how to graph the inverse Determine from a table if the functions are inverses Determine from a graph if the functions are inverses