Opposite Backwards Undo T HE A LGORITHM Starting with a number, - PowerPoint PPT Presentation

D AY 27 – I NVERSE F UNCTIONS

F OLLOWING D IRECTIONS Suppose you are given the following directions:  From home, go north on Route 23 for 5 miles  Turn east (right) onto Orchard Street  Go to the 3rd traffic light and turn north (left) onto Avon Drive  Tracy’s house is the 5th house on the right. If you start from Tracy’s house, write down the directions to get home.

D IRECTIONS TO H OME  Turn left onto Avon Drive as you leave Tracy’s house  Turn South (right) onto Orchard Street. It will be the 3 rd traffic light.  Turn West (left) onto Route 23  Head South for 5 miles until you reach home.

H OW DID YOU COME UP WITH THE DIRECTIONS TO GET H OME FROM T RACY ’ S ?  Opposite  Backwards  Undo

T HE A LGORITHM  Starting with a number, add 5 to it  Divide the result by 3 The final result is 10. Working backwards knowing this result, find the original number. Show your work.

𝑌 = 10 × 3 − 5 = 25 𝑦 = 25

W RITE A F UNCTION Write a function 𝑔(𝑦) , which when given a number x (the original number) will model the operations given above. 𝑦+5 𝑔 𝑦 = 3

𝑕 𝑦 = Write a function 𝑕(𝑦) , which when given a number x (the final result), will model the backward algorithm that you came up with above. 𝑕 𝑦 = 3𝑦 − 5

U SING THE FUNCTIONS YOU DISCOVERED ABOVE , FILL IN THE TABLE . 𝑦 𝒛 = 𝑔(𝑦) 𝒜 = 𝒉(𝒛) 𝒈(𝒜) 10 𝑧 = 10 + 5 𝑔 10 = 10 + 5 𝑨 = 3 × 5 − 5 = 10 = 5 = 5 3 3 1 𝑧 = 1 + 5 𝑔(1) = 1 + 5 𝑨 = 3 × 2 − 5 = 1 = 2 = 2 3 3 4 𝑧 = 4 + 5 𝑔(4) = 4 + 5 𝑨 = 3 × 3 − 5 = 4 = 3 = 3 3 3 19 𝑧 = 19 + 5 𝑔(𝑦) = 19 + 5 𝑨 = 3 × 8 − 5 = 19 = 8 = 8 3 3

D EFINITION OF I NVERSE In this scenario, 𝒈(𝒚) and 𝒉(𝒚) are inverses of each other because g(x) will undo the actions of 𝒈(𝒚) . Thus, we could write 𝒉(𝒚) as 𝒈 −𝟐 (𝒛) described as “f inverse.”

FLOWCHART 1. Design a flowchart explaining how to find the inverse of a function. 2. Design a flowchart explaining how to find the function if you know its inverse.

B E SURE TO INCLUDE THE FOLLOWING DETAILS :  Answer the question, “Do all functions have an inverse?”  Include at least three examples within your flowchart.  Use appropriate notation and use it correctly.  Use appropriate terminology effectively.

You may find it useful to use Microsoft Word or Glogster. Both of these programs (along with many others) have flowchart clipart built in to their software.

FLOWCHART 1. Design a flowchart explaining how to find the inverse of a function. Find the inverse function of 𝑔(𝑦) = 2𝑦 + 4

To find the inverse function, we need to go backtrack, starting with 𝑦 . Do opposite operation of +4 , and that would be −4 .

Do opposite operation of x2, and that would be /2. The inverse function of f(x) = 2x + 4 would be 𝑔 −1 𝑦 = 𝑦−4 2 .

2. Design a flowchart explaining how to find the function if you know its inverse. Find the function, if its inverse is 𝑔 −1 𝑦 = 𝑦−4 2 . To find the function, we need to go backtrack, starting with 𝑦 .

Do opposite operation of /2, and that would be x2. Do opposite operation of -4, and that would be +4. The function is 𝑔 𝑦 = 2𝑦 + 4