Calculus Inverse Functions Test Questions with detailed solutions.

📚 Quick Study Guide

• 🔍 Definition: A function $f^{-1}(x)$ is the inverse of $f(x)$ if $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$.
• 🧮 Finding the Inverse:
  1. Replace $f(x)$ with $y$.
  2. Swap $x$ and $y$.
  3. Solve for $y$.
  4. Replace $y$ with $f^{-1}(x)$.
• 📈 Horizontal Line Test: A function has an inverse if and only if it passes the horizontal line test (i.e., no horizontal line intersects the graph more than once).
• 📝 Derivatives of Inverse Functions: If $y = f^{-1}(x)$, then $\frac{dy}{dx} = \frac{1}{f'(f^{-1}(x))}$.
• 💡 Domain and Range: The domain of $f(x)$ is the range of $f^{-1}(x)$, and the range of $f(x)$ is the domain of $f^{-1}(x)$.

Practice Quiz

1. What is the inverse of the function $f(x) = 2x + 3$?
   1. $f^{-1}(x) = \frac{x - 3}{2}$
   2. $f^{-1}(x) = \frac{x + 3}{2}$
   3. $f^{-1}(x) = 2x - 3$
   4. $f^{-1}(x) = -2x - 3$


2. Which of the following functions does NOT have an inverse?
   1. $f(x) = x^3$
   2. $f(x) = e^x$
   3. $f(x) = x^2$
   4. $f(x) = x + 5$


3. If $f(x) = x^5 + 2$, what is $f^{-1}(x)$?
   1. $f^{-1}(x) = \sqrt[5]{x} - 2$
   2. $f^{-1}(x) = \sqrt[5]{x - 2}$
   3. $f^{-1}(x) = \sqrt[5]{x + 2}$
   4. $f^{-1}(x) = (x - 2)^5$


4. Given $f(x) = \sin(x)$ for $-\frac{\pi}{2} \le x \le \frac{\pi}{2}$, what is $f^{-1}(x)$?
   1. $f^{-1}(x) = \csc(x)$
   2. $f^{-1}(x) = \arcsin(x)$
   3. $f^{-1}(x) = \arccos(x)$
   4. $f^{-1}(x) = \sec(x)$


5. If $f(x) = \frac{1}{x-1}$, what is the domain of $f^{-1}(x)$?
   1. $x \ne 0$
   2. $x \ne 1$
   3. All real numbers
   4. $x > 0$


6. If $f(2) = 7$ and $f'(2) = 3$, what is $(f^{-1})'(7)$?
   1. $\frac{1}{7}$
   2. $3$
   3. $\frac{1}{3}$
   4. $-3$


7. Find the inverse of $f(x) = \ln(x + 3)$.
   1. $f^{-1}(x) = e^x - 3$
   2. $f^{-1}(x) = e^{x+3}$
   3. $f^{-1}(x) = e^x + 3$
   4. $f^{-1}(x) = \ln(x - 3)$