Inverse Functions: Quick Review

Homework policy

• Homework: not marked unless effort is shown; printability discussed; consistency in completing homework helps understanding.

Inverse Functions: Key Concepts

• Inverse undoes the original function; domain and range swap between f and f^{-1}.
• Two standard methods to find inverse:
  • Method A (interchange): Swap x and y, then solve for y to get f^{-1}(x).
  • Method B (reverse operations): Undo each operation in reverse order to solve for x in terms of y, then rename to f^{-1}(x).
• For linear functions y = m x + b:
  • Inverse is f^{-1}(x) = \frac{x - b}{m}.
  • Slope of the inverse is the reciprocal of the original slope: slope(f^{-1}) = 1/m (sign preserved).
• Reminder: Inverse relation is a function only if original is one-to-one.

Example 1: Cost vs. guests

• Original relation: $C = 200 + 10G$ where G = number of guests.
• Inverse relation: $G = \frac{C - 200}{10}$
• Values:
  • Cost for 80 guests: $C = 200 + 10\cdot80 = 1000$
  • For $C = 950$, number of guests: $G = \frac{950 - 200}{10} = 75$
• Interpretation: Original independent variable is G; inverse expresses G as a function of C.

Example 2: Inverse of a simple linear form (two methods)

• Given linear form example, e.g. $y = 3 - \frac{1}{4}x$
  • Method 1 (reverse the operations): starting from $y = 3 - \frac{1}{4}x$, solve for x:
    $\frac{1}{4}x = 3 - y\quad\Rightarrow\quad x = 12 - 4y$
    Inverse: $f^{-1}(x) = 12 - 4x$
  • Method 2 (interchange variables): start with $x = 3 - \frac{1}{4}y$, solve for y:
    $-\frac{1}{4}y = x - 3\quad\Rightarrow\quad y = 12 - 4x$
    Inverse: $f^{-1}(x) = 12 - 4x$
• Result: both methods yield the same inverse.

Quick general case for a linear function

• If $y = m x + b$, then
  • Inverse: $f^{-1}(x) = \frac{x - b}{m}$
  • Slope relationship: slope(f^{-1}) = 1/m (sign preserved).

Practice prompts

• Given $C = 200 + 10G$, find the inverse expression G(C).
• If $f(x) = 3 - \frac{1}{4}x$, find $f^{-1}(x)$.