We need to find some points, graph, and identify the domain and range for the sine function:
[tex]y=\sin x[/tex]
To find some of the points on the graph, we can choose some values of x and then find the respective values of y:
[tex]\begin{gathered} \frac{x\ldots\ldots\ldots\ldots\ldots\ldots\ldots y}{} \\ 0\ldots\ldots\ldots\ldots\ldots\ldots\ldots0 \\ \frac{\pi}{2}\ldots\ldots\ldots\ldots\ldots\ldots\ldots1 \\ \pi\ldots\ldots\ldots\ldots\ldots\ldots\ldots0 \\ \frac{3\pi}{2}\ldots\ldots\ldots\ldots\ldots\ldots-1 \\ 2\pi\ldots\ldots\ldots\ldots\ldots\ldots\ldots0 \\ -\frac{\pi}{2}\ldots\ldots\ldots\ldots\ldots\ldots-1 \\ -\pi\ldots\ldots\ldots\ldots\ldots\ldots\ldots0 \\ -\frac{3\pi}{2}\ldots\ldots\ldots\ldots\ldots\ldots1 \\ -2\pi\ldots\ldots\ldots\ldots\ldots\ldots\ldots0 \end{gathered}[/tex]
Now, we can plot those points and join them smoothly to obtain the graph:
The graph continues indefinity for infinitely negative and positive values of x. Thus, the domain of this function is:
[tex]\begin{gathered} \text{domain: }(-\infty,\infty) \\ or \\ \mleft\lbrace x\in\mathfrak{\Re }\mright\rbrace \end{gathered}[/tex]
And since y can only have values between -1 and 1 (including -1 and 1), we have:
[tex]\begin{gathered} \text{range: }\lbrack-1,1\rbrack \\ or \\ \mleft\lbrace \mright?y\in\mathfrak{\Re }|-1\le y\le1\} \end{gathered}[/tex]
