Algebra 2 test questions on graphing inverse functions and y=x reflection

📚 Quick Study Guide

• 🔍 Inverse Function: An inverse function reverses the operation of the original function. If $f(x)$ maps $x$ to $y$, then $f^{-1}(x)$ maps $y$ back to $x$.
• 🧮 Finding Inverse Functions:
  1. Replace $f(x)$ with $y$.
  2. Swap $x$ and $y$.
  3. Solve for $y$.
  4. Replace $y$ with $f^{-1}(x)$.
• 📈 Graphing Inverse Functions: The graph of an inverse function $f^{-1}(x)$ is the reflection of the graph of the original function $f(x)$ over the line $y = x$.
• 💡 $y = x$ Reflection: Reflecting a point $(a, b)$ over the line $y = x$ results in the point $(b, a)$. This is because the $x$ and $y$ coordinates are swapped.
• 📐 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 function if and only if no horizontal line intersects its graph more than once.

🧪 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 point is the reflection of $(2, 5)$ over the line $y = x$?
   1. $(5, 2)$
   2. $(-2, -5)$
   3. $(2, -5)$
   4. $(-5, -2)$
3. If the range of $f(x)$ is $[1, \infty)$, what is the domain of $f^{-1}(x)$?
   1. $[1, \infty)$
   2. $(-\infty, 1]$
   3. $(0, \infty)$
   4. $(-\infty, 0)$
4. Which of the following functions does NOT have an inverse?
   1. $f(x) = x^3$
   2. $f(x) = x^2$
   3. $f(x) = x + 5$
   4. $f(x) = e^x$
5. What is the domain of the inverse of $f(x) = \sqrt{x - 4}$?
   1. $[0, \infty)$
   2. $[4, \infty)$
   3. $(-\infty, 0]$
   4. $(-\infty, 4]$
6. The graph of $f(x) = 3x - 1$ is reflected over the line $y = x$. What is the equation of the reflected graph?
   1. $y = \frac{x + 1}{3}$
   2. $y = \frac{x - 1}{3}$
   3. $y = -3x + 1$
   4. $y = 3x + 1$
7. Given $f(x) = \frac{1}{x}$, what is $f^{-1}(x)$?
   1. $f^{-1}(x) = \frac{1}{x}$
   2. $f^{-1}(x) = x$
   3. $f^{-1}(x) = -x$
   4. $f^{-1}(x) = x + 1$