Inverses and Algebraic Methods for Finding Them

SEC 6.1.3 What Can I do with Inverses? Page 272

FINDING INVERSES AND JUSTIFYING ALGEBRAICALLY

#6-38 A X Y Consider the table at the right 1 -5 3 7 5 19 7 31 Write an equation for the relationship represented on the table. y=6x-11

#6-38 B X Y Consider the table at the right 1 -5 3 7 5 19 7 31 X Y Make a table for the inverse

6-38C c. How are these two tables related to each other?

6-38D d. Use this relationship to find a shortcut for changing the equation of the original function into its inverse. ? 1? =?+11 6

6-38E e. Now solve this new equation for y.

6-38 F f.Justify that the equations are inverses of each other using any method.

6-39 Find the inverse of the following functions using your new algebraic method, clearly showing all your steps.

6-39A ? = 2(? 1)3 3 ? 2+1 ? 1=

6-39 B ? = ? 1= (? 3)2 ? 2 + 3

6-39 C ? 9 2 ? = 3 + 20 ? 20 3 ? 1(?) = 2 + 9

6-39 D ? =4 3(? 1)3+ 6 ? ?+1 ? ?? = (? ?? ? )

6-40 Adriena's strategy for checking that the functions f(x) and g(x) are inverses is to think of them as stacked function machines. She starts by choosing an input to drop into f(x). Then she drops the output from f(x) into g(x) . If she gets her original number, she is pretty sure that the two equations are inverses.

6-40 A AND B a. Is Adriena's strategy sufficient? Is there anything else she should test to be sure? b. With your team, select a pair of inverse equations from problem 6 39, name them f(x) and g(x), and then use Adriena's ideas to test them.

6-40 C c. Adriena wants to find a shortcut to show her work. She knows if she chooses her input for f(x) to be 3, she can write the output as f(3) . Next, f(3) becomes the input for g(x) , and her output is 3. Since f(3) is the new input for g(x) , she thinks that she can write this process as g(f(3)) . Does her idea make sense? Why or why not?

6-40 D &E d. Her friend, Cemetra thinks she could also write g(f(3)) . Is Cemetra correct? Why or why not. e. Will this strategy for testing inverses work with any input? Choose a variable to use as an input to test with your team's functions, f(x) and g(x).

6-41 Christian, Adriena's teammate, is always looking for shortcuts. He thinks he has a way to adapt Adriena's strategy, but wants to check with his team before he tries it."If I use her strategy but instead of using a number, I skip a step and put the expression f(x) directly into g(x) to create g(f(x)) , will I still be able to show that the equations are inverses?"

6-41 a. What do you think about Christian's changes? What can you expect to get out? a) b) b. Try Christian's idea on your team's equations, and c. Describe your results. c) d) d. Does Christian's strategy shows that the two equations are inverses? How?

6-42 Adriena was finding inverses of some equations. Use your new algebraic strategy to check Adriena's work and test if each pair of equations are inverses of each other. If they are not, explain what went wrong and show how to get the inverse correctly.

6-42 A ? ? =? ? ? =? ?? ?? ?? + ?? Answer: yes

6-42 B ? ? =2(? + 6) + 10 3 ? ? =3 2x 21 Answer: yes

6-42 C ? ? =(? ??)? ? ? ? = ? ? + ?? Answer: NO

6-43 Make a personal poster that shows what you have learned about inverses so far. Create an equation and its inverse then justify that your equations are inverses of each other using several representations.

Review your notes. Rewrite Review and Preview Page 274 # 44-53 ON YOUR OWN: and fortify them if needed. Update your vocab list, if needed.