Solved examples: Graphing inverse functions and their y=x symmetry

📚 Quick Study Guide

• 🔄 Inverse Function Definition: A function that "reverses" another function. If $f(a) = b$, then $f^{-1}(b) = a$.
• 📈 Graphing Inverse Functions: To graph the inverse of a function, swap the $x$ and $y$ coordinates of the original function.
• 🪞 Symmetry about $y=x$: The graph of a function and its inverse are symmetric about the line $y=x$. This means if you were to fold the graph along the line $y=x$, the function and its inverse would overlap.
• ✏️ Finding the Inverse Algebraically:
  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 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. $A) f^{-1}(x) = \frac{x-3}{2}$
2. $B) f^{-1}(x) = \frac{x+3}{2}$
3. $C) f^{-1}(x) = 2x - 3$
4. $D) f^{-1}(x) = -2x - 3$
2. Which of the following points would be on the inverse of $f(x)$ if the point $(2, 5)$ is on $f(x)$?
1. $A) (5, 2)$
2. $B) (2, -5)$
3. $C) (-2, 5)$
4. $D) (-5, -2)$
3. The graph of $f(x)$ is symmetric to the graph of $f^{-1}(x)$ with respect to which line?
1. $A) y = -x$
2. $B) x = 0$
3. $C) y = x$
4. $D) y = 0$
4. If $f(x) = x^2$ for $x \geq 0$, what is $f^{-1}(x)$?
1. $A) f^{-1}(x) = \sqrt{x}$
2. $B) f^{-1}(x) = -\sqrt{x}$
3. $C) f^{-1}(x) = x$
4. $D) f^{-1}(x) = \frac{1}{x^2}$
5. Which function does NOT have an inverse function?
1. $A) f(x) = x + 5$
2. $B) f(x) = 2x$
3. $C) f(x) = x^2$
4. $D) f(x) = x^3$
6. Given the function $g(x) = \frac{1}{x-1}$, find its inverse $g^{-1}(x)$.
1. $A) g^{-1}(x) = \frac{1}{x} + 1$
2. $B) g^{-1}(x) = \frac{1}{x+1}$
3. $C) g^{-1}(x) = x - 1$
4. $D) g^{-1}(x) = 1 - \frac{1}{x}$
7. Which of the following transformations maps $f(x)$ to its inverse?
1. $A) Reflection over the x-axis
2. $B) Reflection over the y-axis
3. $C) Reflection over the line y = x
4. $D) Rotation of 90 degrees