What are Inverse Functions? A Clear Definition and Examples

📚 Quick Study Guide

• 🔄 Definition: An inverse function, denoted as $f^{-1}(x)$, "undoes" the action of the original function $f(x)$. If $f(a) = b$, then $f^{-1}(b) = a$.
• 🧮 Notation: The inverse of $f(x)$ is written as $f^{-1}(x)$. Note that the '-1' is not an exponent; it indicates the inverse.
• 📝 Finding Inverse Functions:
  1. Replace $f(x)$ with $y$.
  2. Swap $x$ and $y$.
  3. Solve for $y$.
  4. Replace $y$ with $f^{-1}(x)$.
• 📈 Graphical Representation: The graph of $f^{-1}(x)$ is a reflection of the graph of $f(x)$ across the line $y = x$.
• ✅ Verification: To verify that two functions are inverses, check if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
• ⚠️ Important Note: Not all functions have an inverse. For a function to have an inverse, it must be one-to-one (i.e., pass the horizontal line test).

Practice Quiz

1. What is the primary characteristic of inverse functions?
   1. They perform the same operation as the original function.
   2. They "undo" the operation of the original function.
   3. They always result in a positive value.
   4. They double the output of the original function.
2. Which of the following is the correct notation for the inverse of the function $f(x)$?
   1. $[f(x)]^{-1}$
   2. $-f(x)$
   3. $f^{-1}(x)$
   4. $\frac{1}{f(x)}$
3. If $f(x) = 2x + 3$, what is $f^{-1}(x)$?
   1. $f^{-1}(x) = \frac{x - 3}{2}$
   2. $f^{-1}(x) = \frac{x + 3}{2}$
   3. $f^{-1}(x) = 2x - 3$
   4. $f^{-1}(x) = -2x - 3$
4. The graph of an inverse function is a reflection of the original function across which line?
   1. $x = 0$
   2. $y = 0$
   3. $y = x$
   4. $y = -x$
5. Which condition must be met for a function to have an inverse?
   1. It must be a quadratic function.
   2. It must be a linear function.
   3. It must be one-to-one.
   4. It must be a many-to-one function.
6. Given $f(x) = x^3$, what is $f^{-1}(8)$?
   1. 2
   2. 4
   3. 64
   4. $\frac{1}{2}$
7. If $f(x) = \sqrt{x - 4}$, what is the domain of $f^{-1}(x)$?
   1. $x \geq 0$
   2. $x \geq 4$
   3. $x \leq 0$
   4. All real numbers