The inverse cosine function, denoted as $y = \arccos(x)$ or $y = \cos^{-1}(x)$, provides the angle whose cosine is $x$. Unlike the regular cosine function, the inverse cosine has a restricted domain of $[-1, 1]$ and a range of $[0, \pi]$. This restriction is crucial to ensure that the inverse cosine is indeed a function (i.e., it passes the vertical line test). Graphing the inverse cosine involves reflecting the restricted portion of the cosine function across the line $y = x$.
Understanding the domain and range is essential when working with inverse cosine functions. The graph starts at $(1, 0)$ and ends at $(-1, \pi)$, creating a decreasing function across its entire domain. Transformations, such as shifts and stretches, can also be applied to the inverse cosine function, altering its graph accordingly.
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Inverse Cosine | A. The set of all possible input values for a function. |
| 2. Domain | B. The function that returns the angle whose cosine is a given number. |
| 3. Range | C. A transformation that shifts a graph horizontally or vertically. |
| 4. Transformation | D. The set of all possible output values for a function. |
| 5. Reflection | E. A transformation that flips a graph over a line. |
The inverse cosine function, denoted as $y = \arccos(x)$, has a domain of __________ and a range of __________. The graph of the inverse cosine function is a __________ function. To obtain the inverse cosine graph, reflect the __________ of the cosine function across the line __________. The value of $\arccos(-1)$ is __________.
Explain how the restricted domain of the cosine function is essential for the existence of the inverse cosine function. What would happen if we did not restrict the domain?
Please log in to post your answer.
Log InEarn 2 Points for answering. If your answer is selected as the best, you'll get +20 Points! 🚀