SLIDE 1
DAY 27 – INVERSE FUNCTIONS
SLIDE 2 FOLLOWING DIRECTIONS
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)
Tracy’s house is the 5th house on the right.
If you start from Tracy’s house, write down the directions to get home.
SLIDE 3 DIRECTIONS TO HOME
Turn left onto Avon Drive as you leave Tracy’s
house
Turn South (right) onto Orchard Street. It will be
the 3rd traffic light.
Turn West (left) onto Route 23 Head South for 5 miles until you reach home.
SLIDE 4
HOW DID YOU COME UP WITH THE
DIRECTIONS TO GET HOME FROM TRACY’S?
Opposite Backwards Undo
SLIDE 5 THE ALGORITHM
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
- riginal number. Show your work.
SLIDE 6
𝑌 =
10 × 3 − 5 = 25 𝑦 = 25
SLIDE 7 WRITE A FUNCTION
Write a function 𝑔(𝑦), which when given a number x (the original number) will model the operations given above. 𝑔 𝑦 =
𝑦+5 3
SLIDE 8
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
SLIDE 9 USING THE FUNCTIONS YOU DISCOVERED
ABOVE, FILL IN THE TABLE.
𝑦 𝒛 = 𝑔(𝑦) 𝒜 = 𝒉(𝒛) 𝒈(𝒜) 10 𝑧 = 10 + 5 3 = 5 𝑨 = 3 × 5 − 5 = 10 𝑔 10 = 10 + 5 3 = 5 1 𝑧 = 1 + 5 3 = 2 𝑨 = 3 × 2 − 5 = 1 𝑔(1) = 1 + 5 3 = 2 4 𝑧 = 4 + 5 3 = 3 𝑨 = 3 × 3 − 5 = 4 𝑔(4) = 4 + 5 3 = 3 19 𝑧 = 19 + 5 3 = 8 𝑨 = 3 × 8 − 5 = 19 𝑔(𝑦) = 19 + 5 3 = 8
SLIDE 10 DEFINITION OF INVERSE
In this scenario, 𝒈(𝒚) and 𝒉(𝒚) are inverses of each other because g(x) will undo the actions
- f 𝒈(𝒚). Thus, we could write 𝒉(𝒚) as 𝒈−𝟐(𝒛)
described as “f inverse.”
SLIDE 11 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.
SLIDE 12
BE 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.
SLIDE 13
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.
SLIDE 14 FLOWCHART
- 1. Design a flowchart explaining how to
find the inverse of a function. Find the inverse function of 𝑔(𝑦) = 2𝑦 + 4
SLIDE 15
To find the inverse function, we need to go backtrack, starting with 𝑦. Do opposite operation of +4, and that would be −4.
SLIDE 16
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 .
SLIDE 17
- 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.
SLIDE 18
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
