Inverse functions flip input and output, letting us undo what a function does. They're like mathematical opposites, reversing each other's effects. Understanding inverses helps us solve equations and model real-world situations where we need to work backwards.

We can find inverses algebraically by swapping x and y, or graphically by flipping over y=x. Domains and ranges swap too. Not all functions have inverses, so we sometimes need to restrict domains to make them work.

Inverse Functions

Verification of inverse functions

• Definition states two functions $f$ and $g$ are inverses if and only if:
  • Composing $f(g(x))$ equals $x$ for all $x$ values in the domain of $g$
  • Composing $g(f(x))$ equals $x$ for all $x$ values in the domain of $f$
• Verify inverse functions by composing $f(g(x))$ and $g(f(x))$
  • If both compositions simplify to $x$, the functions are confirmed inverses
  • Example: Given $f(x) = 2x + 1$ and $g(x) = \frac{x-1}{2}$, verify they are inverses
• Evaluate inverse functions by finding the $x$ value that makes $f(x) = a$
  • Example: If $f(x) = x^2$ and $f^{-1}(9) = 3$, then $f(3) = 9$
• Function notation (e.g., $f(x)$ and $f^{-1}(x)$) is used to represent a function and its inverse

Domains and ranges of inverses

• For a one-to-one function $f$ with domain $A$ and range $B$:
  • The inverse function $f^{-1}$ has domain $B$ and range $A$
  • Domain and range are swapped for inverse functions
• Some functions require domain restrictions to have an inverse
  • Restrict domain to make the function one-to-one
  • Square root functions: Restrict domain to $x \geq 0$
  • Logarithmic functions: Restrict domain to $x > 0$
  • Example: $f(x) = x^2$ is not one-to-one, but $f(x) = x^2$ with $x \geq 0$ is one-to-one and has an inverse

Finding inverse functions

• Algebraically find inverse functions:
  1. Start with $y = f(x)$
  2. Swap $x$ and $y$ to get $x = f(y)$
  3. Solve the equation for $y$ in terms of $x$
  4. Replace $y$ with $f^{-1}(x)$ to get the inverse function
  • Example: Find the inverse of $f(x) = 3x - 2$
• Graphically find inverse functions by reflecting the graph of $f(x)$ over the line $y = x$
  • The reflected graph is the graph of $f^{-1}(x)$
  • Example: The inverse of $f(x) = 2^x$ is $f^{-1}(x) = \log_2 x$, found by reflecting $f(x)$ over $y = x$

Graphing inverses by reflection

• The line $y = x$ acts as a reflection line for the graphs of inverse functions
  • If $(a, b)$ is a point on $f(x)$, then $(b, a)$ is a point on $f^{-1}(x)$
  • Coordinates are swapped when reflecting over $y = x$
• To graph inverse functions:
  1. Graph the original function $f(x)$
  2. Reflect each point of $f(x)$ over the line $y = x$
  3. The reflected points make up the graph of $f^{-1}(x)$
  • Example: The graphs of $f(x) = 2x$ and $f^{-1}(x) = \frac{x}{2}$ are reflections over $y = x$
• This reflection demonstrates the symmetry between a function and its inverse

Properties of Inverse Functions

• Bijection: A function must be both injective (one-to-one) and surjective (onto) to have an inverse
• Monotonicity: Strictly increasing or decreasing functions always have inverses
• Inverse trigonometric functions (e.g., arcsin, arccos, arctan) are examples of functions with restricted domains to ensure invertibility