📚 Topic Summary
The arccosine function, denoted as $y = \arccos(x)$, is the inverse of the cosine function. This means that if $\cos(y) = x$, then $y = \arccos(x)$. The domain of $\arccos(x)$ is $-1 \leq x \leq 1$, and its range is $0 \leq y \leq \pi$. To plot the graph, we essentially reflect the graph of $\cos(x)$ over the line $y=x$, but we only consider the portion of the cosine function that passes the horizontal line test on the interval $[0, \pi]$. Understanding the restricted domain and range is crucial for accurately graphing the arccosine function.
When plotting $y = \arccos(x)$, remember that you're finding the angle (in radians) whose cosine is $x$. The graph starts at the point $(-1, \pi)$, decreases steadily, and ends at the point $(1, 0)$. It's a decreasing function across its entire domain. This activity will guide you through the process, reinforcing your understanding through vocabulary, fill-in-the-blanks, and critical thinking exercises.
🧮 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Arccosine |
A. The set of all possible output values of a function. |
| 2. Domain |
B. The inverse function of the cosine function. |
| 3. Range |
C. A line about which a graph is symmetric. |
| 4. Inverse Function |
D. The set of all possible input values of a function. |
| 5. Symmetry |
E. A function that "reverses" another function. |
✍️ Part B: Fill in the Blanks
Complete the following paragraph using the words provided: inverse, cosine, radians, decreasing, $[-1, 1]$.
The arccosine function is the ________ of the ________ function. Its domain is ________, and its range is $[0, \pi]$ in ________. The graph of $y = \arccos(x)$ is ________ over its entire domain.
🤔 Part C: Critical Thinking
Explain why the domain of $y = \arccos(x)$ is restricted to $[-1, 1]$. What would happen if we tried to find $\arccos(2)$?