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.