INVERSES f -1 (x). Some of the content taken from http://web.psjaisd.us/auston.cron/ABCronPortal/GeoGebraMenu/GeogebraFiles/InverseFunctions/findInvFunc.html
1.
Go to www.tinyurl.com/mthsfisherinverse
2.
In the Input bar, type f(x) = x^3
3.
Use the Point Tool to place a Point A anywhere on the graph of f(x)
4.
In the Input bar, type B=(y(A),x(A)).
B is now considered to be A -1 or A inverse
5.
Use the Drag tool (upper left), and move A along the graph of f(x)
6.
Write down the 3 different coordinates of A (in quadrants 1 and 3) and coordinates of A -1 (B)
A:___________ A -1 :___________
A:___________ A -1 :___________
A:___________ A -1 :___________
7.
What do you notice about A and A -1
8.
Move A to (1,1). What is A -1 ?______________ Move A to (0,0). What is A -1 ?____________
9.
In the Input bar, type f(x) = x^2.
10.
Drag A around and describe how A -1 moves.
11.
Write down the 3 different coordinates of A (in quadrants 1 and 2) and coordinates of A -1 (B)
A:___________ A -1 :___________
A:___________ A -1 :___________
A:___________ A -1 :___________
12.
Graph f -1 (x) on the grid to the right (hint: plot the points of B)
13.
Suppose that f(x) is defined by the relation of {(2,3),(-1,4), and (7,6)}, what would the relation of f -1 (x) be?
______________________________________________________________
14.
Click on the Locusttool . Then click on B and then click on A. You will see the inverse of f -1 (x) appear. How does it compare to your answer to #11?
_______________________________________________
15.
Which of the following lines are f(x) and f -1 (x) reflected on?
Y = 0? x = 0? y= x?
16.
At what point(s) do f(x) and f -1 (x) intersect at?
17.
Is f -1 (x) a function? Explain. _____________________________________
18.
In the Input bar, type f(x) = x 3 – 5.
19.
What line is f -1 (x) reflected on?
______________________________________________
20.
Go to the Grab tool. Click on the y=x line to verify your answer
21.
At what points do f(x) and f -1 (x) intersect at?
22.
Is f -1 (x) a function? Explain________________________________________
23.
Click on Horizontal Line. Move it up and down.
Is there any horizontal that touches f(x) more than once?_______________
24.
How can you tell by looking at f(x) if f -1 (x) is a function?
_______________________________________________

25.
Graph f(x) = 3x – 6. What is the equation of the line of f -1 (x)? _______________________
26.
Graph f(x) = 2x + 8. What is the equation of the line of f -1 (x)?________________________
27.
Suppose you’re at a starting spot. You walk 3 blocks North and then 2 blocks EAST. To return to the same spot going the same route, you would walk 2 blocks WEST and then 3 blocks South.
Suppose you’re at a different starting point, and you walk 4 blocks West and then 2 blocks South. How would you return? _________________________________
28.
Suppose you start with 5. You multiply by two and then add 4 to get 14. To return to 5, you would subtract 4 and then divide by 2. Suppose you start with 6. You divide by 3 and then add 7 to get ______. To return to 6, you would _________________________________________________.
29.
Suppose you start with 4 and you square it then add 3 to get 19. To return to 4, you would __________
_________________________________________________.
30.
Suppose you start with x. You multiply by 3 and then subtract 6. To return to x, you would ___________
________________________________.
31.
Suppose you start with x. You multiply by 2 and then add 8. To return to x, you would ________________
________________________________.
32.
Verify that your answers to 30 and 31 are the same to 25 and 26.
33.
f(x) = ½ x + 4. Find f -1 (x) in a similar method to 31. Verify by graphing f(x) on the computer.
34.
Here’s another way to find f -1 (x) if f(x) = 3x - 6. Describe each step… y = 3x – 6 x = 3y – 6 x + 6 = 3y
1/3x + 2 = y f -1 (x) = (1/3)x + 2
35.
Find the inverse to f(x)= 1/4x - 6 in a similar method to 34. Describe each step.
36.
Find the inverse to f(x) = x 3 + 5. Describe each step. Verify answer with teacher and graphically.
37.
Let f(x) = 3x -6 and g(x) = 1/3x + 2. Find f o g (x) and g o f(x)
What do you notice about f o g(x) and g o f(x)?

38.
Let f(x) = x 3 + 2. g(x) = 3 x

2 . Find f(g(x)) and g(f(x)). Is g(x) = f -1 (x)
39.
f(x) = 2x + 5. g(x) = 1/2x – 5. Find f(g(x)). Is g(x) = f -1 (x)?
Summarize the math.
1.
f(x) = {(a,b),(c,d)}, f -1 ={( , ),( , )}
2.
To find f -1 (x) graphically, you can find coordinates (a,b),(c,d) etc… and change them to ( , ), ( , )
3.
To find f -1 (x) algebraically, switch ____ and _____; then solve for the new ____.
4.
f(x) and it’s inverse will always reflect on the line:
5.
f(x) and f -1 (x) will always intersect at (a,___), (b,___ ) etc…
6.
f o f -1 (x) and f -1 o f(x) will always be simplified to _________
7.
f -1 (x) will be a function if f(x) passes the _______________ line test
Classwork (on separate paper)
1.
Graph f -1 (x) 
2-7. Find f -1 (x)
2.
{(3,-2), (-1,4)}3. f(x) = (x + 2) 3 4. f(x) = 4x – 5 5. f(x) = x 2 + 1 (x>0) 6. F(x) = x
1

2 x
7.f(x) =
1
 x
8. Prove that your answer to #4 is true by showing the graph and doing the composition.
9.
f(x) = 2x + 6. g(x) = 0.5x – 6. Are f(x) and g(x) inverses? Prove it two ways.
10.
Jane has a 20% off coupon and then has to pay 7% sales tax. a.
Write two functions that show the discount, d(x), and tax, t(x). b.
Which shows her final price, F(x), d o t(x) or t o d(x)? Explain. c.
Find F(x) and what does it represent? d.
Find F -1 (x). What does it represent?
11.
Most foreign currencies (even the obscure ones) are given an exchange rate based on USD. Supposed that
99.3 Yen = 1$ and 13.32 Mexican Pesos = 1$. a.
Write two functions Y(x) and M(x) that calculate the number of Yen and Pesos for each dollar. b.
Find Y -1 (x) and M -1 (x). What do they represent? c.
Suppose that Jane is travelling in Mexico and wants to purchase something for 65 Pesos. Which function would she use to find the cost in USD? d.
Suppose that Tomoko is traveling in Mexico and wants to calculate the cost in Yen. Call this function J(x).
Which of the following composite functions will she use? M o Y -1 (x), Y o M(x), M o Y(x) or Y o M -1 (x)? e.
Find J(x). Verify your answer by calculating the cost of 13.32 pesos to Japanese Yen. f.
Find J -1 (x). Verify your answer by calculating the cost of 993 yen to pesos.