Inverse Engineering Slides Algebra II: Unit 4
Inverse Engineering
Mastering Exponential & Logarithmic Inverses
The Un-Doer
"I am thinking of a number. I multiply it by 3, add 7, and then square the result. My final answer is 100."
The Mission
Can you work backwards to find my original number?
The Connection
What operations did you have to 'undo' and in what order?
Algebraic Methods
Part 1: Solving Inverses
Embedded media
Clip: Exponential to Logarithmic (4:10 - 5:13)
Key Steps
Swap x and y
Isolate the base
Convert form (Log/Exp)
Solve for y
Algebraic Methods
Part 2: Log to Exponential
Embedded media
Clip: Logarithmic to Exponential (7:58 - 8:15)
Pro Tip: Domain & Range
The domain of \( f(x) \) is the range of \( f^{-1}(x) \).
They Swap!
Station Rotation
S1
Exponential Focus
Convert complex exponentials with shifts and reflections into logs.
S2
Logarithmic Focus
Handle the Natural Log (\(\ln\)) and Common Log (\(\log\)) conversions.
S3
Verification Lab
Prove your work using composition: \( f(f^{-1}(x)) = x \).
8 Minutes Per Station
Teams of 3-4
Mission Complete?
Show what you know on the Exit Ticket!
Find the inverse of:
Inverse Engineering Worksheet Inverse Engineering
Station Rotation Challenge
Name:
Date:
1
Station: Exponential to Logarithmic
Level 1: Find the inverse of \( f(x) = 4^{x-3} \)
Level 2: Find the inverse of \( g(x) = 5 - 2(3)^{x+1} \)
2
Station: Logarithmic to Exponential
Level 1: Find the inverse of \( h(x) = \log_2(x+5) \)
Level 2: Find the inverse of \( k(x) = \frac{1}{2}\ln(x-1) + 4 \)
3
Station: Verification Lab
Objective: Verify that \( f(x) = 10^{x+2} \) and \( g(x) = \log(x) - 2 \) are inverses using composition.
Hint: You must show that \( f(g(x)) = x \) AND \( g(f(x)) = x \).
Prove \( f(g(x)) = x \)
Prove \( g(f(x)) = x \)
Inverse Engineering Exit Ticket Exit Ticket
Inverse Engineering Final Check
Technician Name:
Date:
The Challenge
Find the inverse of the following function algebraically:
\( y = 2\ln(x) + 1 \)
Show Your Inverse Engineering Steps Below:
Remember: Isolate the log term before converting to exponential form!
Inverse Engineering Teacher Guide Teacher Blueprint
Lesson: Inverse Engineering Functions
45 Minutes
Learning Objective
Students will algebraically calculate the inverses of exponential and logarithmic functions involving vertical shifts, horizontal shifts, and vertical stretches. They will verify these inverses using function composition.
Lesson Flow
05m
The Un-Doer Hook
"I think of a number..." See slides for prompt. Reversing operations is the core of inverses.
10m
Direct Modeling (Video)
Play segments 4:10-5:13 and 7:58-8:15. Pause for student questions on the conversion step.
25m
Station Rotation
3 stations, ~8 mins each. Students rotate through Exp Focus, Log Focus, and Verification Lab.
05m
Exit Ticket
Individual assessment of natural log inverse calculation.
Materials Needed
Slides & Projector
Student Worksheets
Exit Tickets
Station Timer
Key Vocabulary
Composition: \( f(g(x)) = x \)
Inverse: The "un-doer" function
Isolation: Crucial first step before log/exp conversion
Answer Key & Teacher Tips
Worksheet Solutions
Station 1 - Level 1:
\( f^{-1}(x) = \log_4(x) + 3 \)
Station 1 - Level 2:
\( g^{-1}(x) = \log_3\left(\frac{x-5}{-2}\right) - 1 \)
Station 2 - Level 1:
\( h^{-1}(x) = 2^x - 5 \)
Station 2 - Level 2:
\( k^{-1}(x) = e^{2(x-4)} + 1 \)
Exit Ticket Solution
1. \( x = 2\ln(y) + 1 \)
2. \( x - 1 = 2\ln(y) \)
3. \( \frac{x-1}{2} = \ln(y) \)
Answer: \( y = e^{\frac{x-1}{2}} \)
Common Misconceptions
Watch out for:
Order of Operations: Students trying to convert to log form before isolating the exponential base.
Swap Errors: Swapping x and y at the very end instead of the start.
Base Confusion: Using the wrong base in the log (e.g., using base 10 for \( e^x \)).
