Lecture Notes: Exam Preparation and Inverse Functions

Exam 2 Overview

• Timing: Occurs right after spring break.
• Material Covered: Unit two, excluding problematic parts of section 3.6.
• Preparation Tips:
  • Use GradeScope for exam submission.
  • Print templates as PDFs, not Google Docs.
• Key Focus Areas:
  • Problems on graph transformation and vertical asymptotes.
  • Quadratic functions: vertex form, intercepts, graph sketching.
  • Zeros and multiplicity: end behavior, synthetic and long division.
  • Intermediate value theorem, domain, and asymptotes.
  • Inequalities and rational powers.

Class Structure

• Practice Exams: Available mostly for terminal classes like statistics and contemporary math.
• Upcoming Classes: Regular classes continue on Wednesdays.
• Homework: Lighter assignment due April 2nd, covering inverse functions.

New Material: Inverse Functions

Inverse Function Concepts

• Invertible Functions: Can be reversed (one-to-one functions).
• One-to-One Functions:
  • Each input has a unique output.
  • Pass horizontal line test.

Horizontal Line Test

• To determine if a function is one-to-one:
  • A function is not one-to-one if any horizontal line intersects it more than once.

Inverse Function Notation

• Notated as ( f^{-1}(x) ), not a reciprocal.
• Properties:
  • ( f(f^{-1}(x)) = x )
  • ( f^{-1}(f(x)) = x )

Finding Inverses

• Process:
  1. Replace ( f(x) ) with ( y ).
  2. Swap ( x ) and ( y ).
  3. Solve for ( y ).
  4. Notate as ( f^{-1}(x) ).

Determining Inverses

• Check if two functions are inverses by substituting one into the other and checking if it results in ( x ).

Examples

• Graphical Determination:
  • Use horizontal line test to confirm if a function is one-to-one.
• Table Analysis:
  • Check for unique outputs and non-repeated inputs.
• Inverse Calculation:
  • Apply the process above to solve for the inverse.

Advanced Topics

• Domain and Range in Inverses:
  • Domain and range swap in an inverse function.
  • Ensure correct domain/range for inverse, particularly when given as intervals.

Word Problems

• Understand the meaning of inverses in real-world contexts, e.g., time vs. distance functions.

Conclusion

• Prepare for exams by focusing on highlighted problems.
• Understand and apply concepts of inverse functions and one-to-one properties.
• Continue practicing the preparation and calculation of inverses for future lessons.