Quiz: Do You Understand Inverse Function Properties?

📚 Quick Study Guide

• 🔑 Definition: An inverse function, denoted as $f^{-1}(x)$, "undoes" the original function $f(x)$. In other words, if $f(a) = b$, then $f^{-1}(b) = a$.
• 🔄 Domain and Range: The domain of $f(x)$ becomes the range of $f^{-1}(x)$, and the range of $f(x)$ becomes the domain of $f^{-1}(x)$.
• 📈 Horizontal Line Test: A function has an inverse if and only if it passes the horizontal line test (no horizontal line intersects the graph more than once).
• 🧮 Finding the Inverse: To find the inverse of $f(x)$:
  1. Replace $f(x)$ with $y$.
  2. Swap $x$ and $y$.
  3. Solve for $y$.
  4. Replace $y$ with $f^{-1}(x)$.
• 🧩 Composition Property: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all $x$ in the respective domains.
• отражение Graphical Representation: The graph of $f^{-1}(x)$ is the reflection of the graph of $f(x)$ across the line $y=x$.

🧠 Practice Quiz

1. If $f(2) = 5$, which of the following is true about the inverse function $f^{-1}(x)$?

   1. $f^{-1}(5) = 2$
   2. $f^{-1}(2) = 5$
   3. $f^{-1}(5) = 5$
   4. $f^{-1}(2) = 2$

2. Which of the following functions has an inverse?

   1. $f(x) = x^2$
   2. $f(x) = |x|$
   3. $f(x) = x^3$
   4. $f(x) = \sin(x)$

3. What is the inverse of $f(x) = 2x + 3$?

   1. $f^{-1}(x) = \frac{x - 3}{2}$
   2. $f^{-1}(x) = \frac{x + 3}{2}$
   3. $f^{-1}(x) = \frac{2}{x - 3}$
   4. $f^{-1}(x) = \frac{2}{x + 3}$

4. If $f(x) = x - 4$ and $f^{-1}(x)$ is its inverse, what is $f^{-1}(7)$?

   1. 3
   2. 11
   3. -3
   4. -11

5. Which statement best describes the relationship between the graphs of a function and its inverse?

   1. They are reflections across the x-axis.
   2. They are reflections across the y-axis.
   3. They are reflections across the line $y = x$.
   4. They are translations of each other.

6. If $f(x) = \frac{1}{x}$, what is $f^{-1}(x)$?

   1. $f^{-1}(x) = x$
   2. $f^{-1}(x) = -x$
   3. $f^{-1}(x) = \frac{1}{x}$
   4. $f^{-1}(x) = -\frac{1}{x}$

7. What is the domain of the inverse of $f(x) = \sqrt{x + 2}$?

   1. $x \ge -2$
   2. $x \ge 0$
   3. $x \le -2$
   4. All real numbers