Describe the general shape of the graph for the inverse sine function, $$ y = \sin^{-1}(x) $$, including its domain and range.

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The inverse sine function, denoted as $$ y = \sin^{-1}(x) $$ or $$ y = \arcsin(x) $$, is the inverse of the sine function when the sine function is restricted to its principal branch. Here's a detailed description of its graph, including its general shape, domain, and range:

### **General Shape of the Graph**

1. **Shape and Behavior:**
   - **Increasing Function:** The graph of $$ \sin^{-1}(x) $$ is an **increasing** function. As $$ x $$ increases from -1 to 1, $$ y = \sin^{-1}(x) $$ also increases smoothly.
   - **Points of Interest:**
     - **Origin:** The graph passes through the origin $$(0, 0)$$, since $$ \sin^{-1}(0) = 0 $$.
     - **Endpoints:** 
       - At $$ x = 1 $$, $$ y = \frac{\pi}{2} $$ because $$ \sin^{-1}(1) = \frac{\pi}{2} $$.
       - At $$ x = -1 $$, $$ y = -\frac{\pi}{2} $$ because $$ \sin^{-1}(-1) = -\frac{\pi}{2} $$.
   - **Symmetry:** The graph is **odd-symmetric**, meaning it is symmetric with respect to the origin. This implies that $$ \sin^{-1}(-x) = -\sin^{-1}(x) $$.

2. **Curve Characteristics:**
   - The curve is smooth and continuous within its domain.
   - It starts at the point $$(-1, -\frac{\pi}{2})$$, passes through $$(0, 0)$$, and ends at $$(1, \frac{\pi}{2})$$.
   - The graph has a gentle, S-shaped curve, reflecting the behavior of the sine function's inverse.

### **Domain and Range**

1. **Domain:**
   - The domain of $$ y = \sin^{-1}(x) $$ consists of all real numbers $$ x $$ for which the function is defined.
   - **Domain:** $$ x \in [-1, 1] $$
   - This is because the sine function only outputs values between -1 and 1, and thus its inverse can only accept inputs within this interval.

2. **Range:**
   - The range of $$ y = \sin^{-1}(x) $$ consists of all possible output values $$ y $$ that the function can produce.
   - **Range:** $$ y \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] $$
   - This interval represents the principal values for the inverse sine function, ensuring that each input $$ x $$ has a unique corresponding output $$ y $$.

### **Visual Representation**

Here's a summary of key points to visualize the graph:

- **Endpoints:** $$(-1, -\frac{\pi}{2})$$ and $$(1, \frac{\pi}{2})$$
- **Intercept:** $$(0, 0)$$
- **Shape:** Smooth, increasing S-curve
- **Symmetry:** Odd function (symmetric about the origin)

![Graph of the Inverse Sine Function](https://i.imgur.com/5G5bWZT.png)

*Note: The above image is a representative graph of the inverse sine function $$ y = \sin^{-1}(x) $$.*

### **Summary**

- **Function:** $$ y = \sin^{-1}(x) $$
- **Domain:** $$ x \in [-1, 1] $$
- **Range:** $$ y \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] $$
- **Graph Characteristics:** Smooth, continuous, increasing S-shaped curve passing through $$(-1, -\frac{\pi}{2})$$, $$(0, 0)$$, and $$(1, \frac{\pi}{2})$$, with origin symmetry.

This comprehensive description should help you understand the overall appearance and properties of the inverse sine function's graph.
The inverse sine function $$ y = \sin^{-1}(x) $$ is an increasing, smooth, S-shaped curve that passes through the points $$(-1, -\frac{\pi}{2})$$, $$(0, 0)$$, and $$(1, \frac{\pi}{2})$$. Its domain is all real numbers between -1 and 1, and its range is from $$-\frac{\pi}{2}$$ to $$\frac{\pi}{2}$$.

Describe the general shape of the graph for the inverse sine function, \( y = \sin^{-1}(x) \), including its domain and range.

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