Steps to determine if a function has an inverse using the horizontal line test.

📚 Quick Study Guide

• 📈 A function has an inverse if and only if it is a one-to-one function (i.e., it passes both the vertical and horizontal line tests).
• ↔️ The Horizontal Line Test: A function $f(x)$ has an inverse if no horizontal line intersects its graph more than once.
• ✍️ Graph the function. This is crucial for visual inspection.
• 📏 Draw horizontal lines across the graph at different y-values.
• 👁️‍🗨️ If any horizontal line intersects the graph more than once, the function does not have an inverse. If all horizontal lines intersect at most once, the function has an inverse.
• 📝 Mathematically, a function $f$ is one-to-one if for any $x_1$ and $x_2$ in its domain, if $f(x_1) = f(x_2)$, then $x_1 = x_2$.
• 💡 Remember, if a function has an inverse, the graph of the inverse is the reflection of the original function across the line $y = x$.

Practice Quiz

1. Which of the following statements accurately describes the horizontal line test?

   1. A) It determines if a function is continuous.
   2. B) It determines if a function is differentiable.
   3. C) It determines if a function has an inverse.
   4. D) It determines the slope of a function.

2. Consider the function $f(x) = x^2$. Does this function have an inverse over all real numbers?

   1. A) Yes, because it's a polynomial.
   2. B) No, because it fails the horizontal line test.
   3. C) Yes, because it passes the vertical line test.
   4. D) No, because it's not continuous.

3. Which of the following functions has an inverse?

   1. A) $f(x) = |x|$
   2. B) $f(x) = x^3$
   3. C) $f(x) = x^2$
   4. D) $f(x) = \sin(x)$

4. A graph of a function is given. A horizontal line intersects the graph at three points. What can you conclude?

   1. A) The function is not continuous.
   2. B) The function has an inverse.
   3. C) The function does not have an inverse.
   4. D) The function is a straight line.

5. Which function below would pass the horizontal line test?

   1. A) A parabola opening upwards.
   2. B) A cubic function that is always increasing.
   3. C) A circle.
   4. D) A cosine wave.

6. If a function $f(x)$ passes the horizontal line test, what does that indicate about the function?

   1. A) It is decreasing.
   2. B) It is increasing.
   3. C) It is one-to-one.
   4. D) It is discontinuous.

7. The function $f(x) = \sqrt{x}$ is defined for $x \geq 0$. Does this function have an inverse?

   1. A) No, because square roots are always positive.
   2. B) Yes, because it passes the horizontal line test.
   3. C) No, because it fails the vertical line test.
   4. D) Yes, because it passes the vertical line test.