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

2. Graphing Inverses

3. 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

4. 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.

5. 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

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

7. 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

8. 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!!!

9. Lets make a table What do you notice about the two tables? 𝑥 𝒇(𝑥) 1 31 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?

10. 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

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

12. Graphs
What do you notice about the graphs?

13. 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.

14. It should look like this…

15. 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

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

17. 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