https://file.aiquickdraw.com/mathbot/file/1726809526197vcvxd44a.jpg

The graph in the image is a portion of the sine function \( f(x) = \sin(x) \), and the questions ask about the domain and range of the function.

### a. **Domain of the function:**
The domain of the sine function, \( f(x) = \sin(x) \), is all real numbers because the sine function is defined for all \( x \in \mathbb{R} \). In mathematical notation:

\[
\text{Domain of } f(x) = (-\infty, \infty)
\]

### b. **Range of the function:**
The range of the sine function is based on the fact that the sine of any real number always lies between -1 and 1, inclusive. From the graph, this is also visually confirmed as the maximum and minimum values the function attains are 1 and -1, respectively. Hence, the range is:

\[
\text{Range of } f(x) = [-1, 1]
\]

Would you like further details or clarifications?

Here are 5 related questions to expand your understanding:
1. What is the period of the sine function \( \sin(x) \)?
2. How would the range change if the function were \( f(x) = 2\sin(x) \)?
3. How does the domain change if the sine function is shifted horizontally?
4. What is the amplitude of \( f(x) = \sin(x) \)?
5. How can the sine function be used to model real-world periodic phenomena?

**Tip:** The period of \( f(x) = \sin(x) \) is \( 2\pi \), which means it repeats its shape every \( 2\pi \) units along the x-axis.

A portion of the graph of f(x) = sin(x) is shown. a. What is the domain of the function? b. What is the range of the function?

Explore the domain and range of the sine function f(x) = sin(x). This problem involves key trigonometric concepts, including how the function is defined for all real numbers and its range between -1 and 1. Suitable for high school students in Grades 9-12.

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