Question

graph the function ( y = 3sinleft(\frac{1}{3}x
ight) ). show at least two cycles. use the graph to determine the domain and range of the function.
use the graphing tool to graph the equation. type pi to insert ( pi ) as needed.
use the graph to determine the domain of ( y = 3sinleft(\frac{1}{3}x
ight) ).
(type your answer in interval notation. use integers or fractions for any numbers in the expression.)
use the graph to determine the range of ( y = 3sinleft(\frac{1}{3}x
ight) ).
(type your answer in interval notation. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the domain of sine function

The general form of a sine function is \(y = A\sin(Bx - C)+D\). For the function \(y=\sin(u)\), the domain of \(u\) (in our case \(u = \frac{1}{3}x\)) is all real numbers. Since there are no restrictions on the value of \(x\) for which \(\frac{1}{3}x\) is defined (because for any real - number \(x\), the operation of multiplying \(x\) by \(\frac{1}{3}\) is valid), the domain of \(y = 3\sin(\frac{1}{3}x)\) is the set of all real numbers. In interval notation, the set of all real numbers is \((-\infty,\infty)\).

Step2: Recall the range of sine function

The range of the basic sine function \(y=\sin(t)\) is \([- 1,1]\). For the function \(y = A\sin(Bx - C)+D\), the range is given by \([D - |A|,D + |A|]\). In the function \(y = 3\sin(\frac{1}{3}x)\), we have \(A = 3\), \(B=\frac{1}{3}\), \(C = 0\), and \(D = 0\).
We know that \(-1\leqslant\sin(\frac{1}{3}x)\leqslant1\). Multiply each part of the inequality by \(3\):

$$
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$$

In interval notation, the range is \([-3,3]\).

Answer:

• Domain: \((-\infty,\infty)\)
• Range: \([-3,3]\)