The inverse sine function, denoted as $y = \arcsin(x)$ or $y = \sin^{-1}(x)$, answers the question: "What angle has a sine of x?" Unlike the regular sine function, its domain is restricted to $[-1, 1]$, and its range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. This restriction is crucial for the inverse to be a function (pass the vertical line test). Graphing involves plotting points, understanding the restricted domain and range, and recognizing the reflection of the sine function across the line $y=x$.
Graphing inverse sine functions requires a solid understanding of transformations, including shifts and stretches. Knowing the base graph of $y = \arcsin(x)$ is essential. From there, you can apply transformations such as vertical and horizontal shifts, as well as vertical and horizontal stretches or compressions, to graph more complex variations.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Inverse Sine | A. The set of all possible output values of a function. |
| 2. Domain | B. The function that returns the angle whose sine is a given number. |
| 3. Range | C. A transformation that shifts a graph horizontally or vertically. |
| 4. Transformation | D. The set of all possible input values of a function. |
| 5. Reflection | E. A transformation creating a mirror image across a line. |
The inverse sine function, also known as _________ (1), has a restricted _________ (2) of $[-1, 1]$. Its range is restricted to $[-\frac{\pi}{2}, \frac{\pi}{2}]$ to ensure it is a _________ (3). To graph $y = \arcsin(x)$, you can reflect the graph of $y = \sin(x)$ over the line _________ (4), but only the portion where $x$ is between _________ (5) and _________ (6).
Explain why the domain of the inverse sine function is restricted. What would happen if we didn't restrict the domain, and how would it affect the graph?
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