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.