Solved examples: Graphing the inverse tangent function y = arctan x

📚 Quick Study Guide

• 🤔 Definition: The inverse tangent function, denoted as $y = \arctan x$ or $y = \tan^{-1} x$, gives the angle whose tangent is $x$.
• 📈 Domain: The domain of $y = \arctan x$ is all real numbers, $(-\infty, \infty)$.
• 📉 Range: The range of $y = \arctan x$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$. This is because the principal values for the inverse tangent lie between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.
• 📍 Key Points:
  • $\arctan(0) = 0$
  • $\arctan(1) = \frac{\pi}{4}$
  • $\arctan(-1) = -\frac{\pi}{4}$
• asymptote: The graph of $y = \arctan x$ has horizontal asymptotes at $y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$.
• ✏️ Graph Shape: The graph of $y = \arctan x$ is increasing and has a point of inflection at $(0, 0)$. It approaches the horizontal asymptotes as $x$ goes to $\pm \infty$.

🧪 Practice Quiz

1. What is the domain of the function $y = \arctan x$?
1. (A) $(0, \infty)$
2. (B) $(-\infty, \infty)$
3. (C) $[-1, 1]$
4. (D) $(-\frac{\pi}{2}, \frac{\pi}{2})$
2. What is the range of the function $y = \arctan x$?
1. (A) $(0, \infty)$
2. (B) $(-\infty, \infty)$
3. (C) $[-1, 1]$
4. (D) $(-\frac{\pi}{2}, \frac{\pi}{2})$
3. What is the value of $\arctan(1)$?
1. (A) $0$
2. (B) $\frac{\pi}{2}$
3. (C) $\frac{\pi}{4}$
4. (D) $\pi$
4. What is the value of $\arctan(0)$?
1. (A) $0$
2. (B) $\frac{\pi}{2}$
3. (C) $\frac{\pi}{4}$
4. (D) $\pi$
5. As $x$ approaches infinity, what value does $\arctan(x)$ approach?
1. (A) $0$
2. (B) $\infty$
3. (C) $\frac{\pi}{2}$
4. (D) $-\frac{\pi}{2}$
6. As $x$ approaches negative infinity, what value does $\arctan(x)$ approach?
1. (A) $0$
2. (B) $\infty$
3. (C) $\frac{\pi}{2}$
4. (D) $-\frac{\pi}{2}$
7. Which of the following statements is true about the graph of $y = \arctan x$?
1. (A) It is decreasing.
2. (B) It has a vertical asymptote at $x = 0$.
3. (C) It passes through the point $(1, 0)$.
4. (D) It is increasing.