Inverse Workshop Slides Tier 2 Intervention
REVERSE
GEAR
Mastering Inverse Functions
Undo Logic
y = x Reflection
The "Undo" Button
What does an inverse actually do?
1 Original Function \(f(x)\)
Takes an Input (x) and turns it into an Output (y).
(x, y)
2 Inverse Function \(f^{-1}(x)\)
Takes that Output (y) and sends it back to the Original Input (x).
(y, x)
Summary: We just swap the roles of \(x\) and \(y\)!
The Reflection Rule
Geometry of Inverses
The Mirror Line
The inverse of a function is a reflection over the line y = x.
Why \(y = x\)?
Because that is the line where the x-values and y-values are exactly the same. Crossing it literally "swaps" them.
Tracing Paper Hack: Draw the function, fold at the 45° angle!
y = x
f(x)
f⁻¹(x)
The Algebraic Recipe
1
REPLACE
Swap the fancy function notation for a simple variable.
f(x) → y
2
SWAP
Switch the position of every \(x\) and every \(y\).
(x) ↔ (y)
3
SOLVE
Isolate the new \(y\) using inverse operations.
y = [...]
Final Step:
y → f⁻¹(x)
Watch Your Step
Trap #1: Reciprocal vs. Inverse
The \( -1 \) is NOT an exponent. It is a label.
f⁻¹(x) = 1/f(x)
It's a Name!
Trap #2: Solving Order
When solving for the new \(y\), go Reverse PEMDAS (SADMEP).
Undo addition/subtraction BEFORE multiplication/division.
Inverse Blueprint Worksheet INVERSE BLUEPRINT
Small Group Intervention | HS.F-BF.B.4
Name:
Date:
1
The Reflection Test
Instructions: Use your sheet of tracing paper.
Trace the axis and the diagonal line \(y = x\).
Trace the solid line \(f(x)\).
Flip the paper diagonally along the \(y = x\) line.
Draw where the new line landed—that is your inverse!
y = x
f(x)
Sketch \(f^{-1}(x)\) above
Coordinate Swap
Original (x, y) Inverse (y, x) (0, 2) (1, 4) (-1, 0)
Hint: To graph the inverse, just plot these new coordinates!
2
The Algebra Blueprint
Follow the scaffolds to solve for the inverse of \(f(x) = 3x - 6\).
Step 1
Replace \(f(x)\) with \(y\)
y = 3x - 6
Step 2
Swap \(x\) and \(y\)
Step 3
Solve for the new \(y\)
Remember: Undo Subtraction first!
y =
Inverse Progress Check PROGRESS CHECK
Inverse Function Proficiency
___ / 3
Student Name
Date
1
Find the inverse of the linear function:
f(x) = 5x - 10
Work Area:
f⁻¹(x) =
2
Find the inverse of the rational function:
g(x) = \(\frac{x + 4}{2}\)
Work Area:
g⁻¹(x) =
3
Points of Reflection:
If the function \(h(x)\) contains the point ( -3 , 7 ), what point must be on the graph of the inverse function \(h^{-1}(x)\)?
(
,
)
Standard HS.F-BF.B.4 • Reflection Mastery
Inverse Facilitator Guide Teacher Guide
Inverse Function Workshop
TIER 2 INTERVENTION
CO Standard: HS.F-BF.B.4
Objective
Students will understand inverse functions as reflections over the line \(y=x\) and master the algebraic "swap and solve" method for linear and basic rational functions.
Prerequisites
Solving multi-step linear equations
Coordinate plane graphing
Function notation basics
Materials Needed
Tracing paper (1 sheet/student)
Coloured pencils (Blue/Red/Green)
Rulers
Pacing
Direct Instruction 10m
Tracing Activity 10m
Guided Algebra 15m
Progress Monitoring 10m
Intervention Strategies
1
Visual Reflection
The tracing paper activity is crucial for Tier 2 learners who struggle with abstract notation. By physically folding the paper, they internalize the \(y=x\) symmetry.
2
Reverse PEMDAS (SADMEP)
Reinforce that isolating \(y\) in Step 3 requires undoing operations in reverse order. Use the "Wrapping a Gift" analogy: Inverse is the "Unwrapping" process.
Misconception Alert
Students often think \(f^{-1}(x)\) means \(\frac{1}{f(x)}\).
INCORRECT: Inverse of 2x+1 is 1/(2x+1)
Master Answer Key
Blueprint Worksheet
Coordinate Swap: (2, 0), (4, 1), (0, -1)
Algebra 3x-6: \(f^{-1}(x) = \frac{x + 6}{3}\)
Progress Monitoring (Exit Ticket)
Problem 1
\(f^{-1}(x) = \frac{x + 10}{5}\)
Problem 2
\(g^{-1}(x) = 2x - 4\)
Problem 3
( 7 , -3 )
