Inverse Functions, Exponentials, and Logarithms

Inverse Functions

  • Definition: The inverse of a function interchanges the domain and range.
    • If f contains the point (a,b), then f^{-1} contains (b,a).
    • When ext{Dom}(f)={\text{all real numbers}} and ext{Range}(f)={\text{positive real numbers}}, then ext{Dom}(f^{-1})={\text{positive real numbers}} and ext{Range}(f^{-1})={\text{all real numbers}}.
  • Graphical Relationship
    • Obtain f^{-1} by switching the x and y coordinates of every ordered pair on f.
    • The graphs of f and f^{-1} are reflections of one another across the line y=x.
  • Line-Tests
    • Vertical Line Test (VLT): determines if a relation is a function.
    • Horizontal Line Test (HLT): determines if the inverse of a function is itself a function.
    • If any horizontal line intersects f more than once, f^{-1} will fail the VLT and will not be a function.
    • Functions that are strictly increasing or strictly decreasing pass the HLT, so their inverses are also functions.

Worked Graph Example: f(x)=x^2\,\text{ for }x\ge0

  • Table of representative points on f: (0,0),(1,1),(2,4),(3,9),(4,16),(5,25).
  • Switching coordinates gives the inverse points: (0,0),(1,1),(4,2),(9,3),(16,4),(25,5).
  • The inverse curve is the right branch of y=\sqrt{x} (only $x\ge0$ because the original domain was x\ge0).
  • HLT passes (each horizontal line hits f once), so f^{-1} is a function.

Exponential Functions and Their Inverses

  • Prototype Activity: f(x)=2^x
    1. Build a table for f:
    • (-2,\tfrac14),(-1,\tfrac12),(0,1),(1,2),(2,4),(3,8).
    1. Plot points and draw the smooth exponential curve.
    2. Draw y=x, reflect each point to form f^{-1}.
    3. The reflected points generate a new curve that passes the VLT; thus the inverse of an exponential function is itself a function.

Logarithms: the Inverse of Exponentials

  • Notation: y=\log_b(x)\iff x=b^y, where b>0 and b\neq1.
    • Read "log base b of x."
    • The logarithm returns an exponent.
  • Equivalence Examples
    • 2^x=8\;\Longleftrightarrow\;x=\log_2(8). Answer: x=3 because 2^3=8.
    • 2^4=16\;\Longleftrightarrow\;\log_2(16)=4.
    • \log_5(25)=2\;\Longleftrightarrow\;5^2=25.
  • Evaluation Example
    • "What is \log_2(32)?"
    • Ask: "2 raised to what power equals 32?"
    • 2^5=32, so \log_2(32)=5.

Attributes of y=\log_b(x) Derived from y=b^x

  1. Graph of y=\log_b(x) is the reflection of y=b^x over y=x.
  2. Domain: x>0 (all positive real numbers).
  3. Range: all real numbers.
  4. x-intercept: (1,0). (Corresponds to y-intercept (0,1) of y=b^x.)
  5. Monotonicity: If y=b^x is increasing (which it is for b>1) or decreasing (for 0<b<1), then \log_b(x) is also respectively increasing or decreasing.
  6. Asymptote: y=b^x has horizontal asymptote y=0; \log_b(x) has vertical asymptote x=0.

Conceptual Take-Aways

  • “Inverse” means switch roles of x and y (domain and range).
  • Reflections over y=x visualize inverse pairs.
  • The exponent in an exponential equation becomes the value of a logarithm.
  • Line Tests Summary:
    • VLT → verifies a relation is a function.
    • HLT → verifies the inverse will also be a function.
  • For logarithms, think "What power of the base equals the argument?"

Miscellaneous Classroom Context & Memory Hooks

  • Students used popcorn and pauses to work problems—reminder that active practice beats passive viewing.
  • Whenever stuck converting between exponential and logarithmic forms, repeat: “Logarithm = exponent.”
  • Common next topics: common logs (\log_{10}) and natural logs (\ln) follow the same rules; only the base changes.