Question

find the domain and range of the function ( y = 3sin(x) ). complete the sentence.
the functions domain is all real numbers and the range is ( ygeq0 )( yleq0 )( -3lt ylt3 )( -3leq yleq3 )( )all real numbers

Explanation:

Step1: Recall the range of \(y = \sin(x)\)

The range of \(y=\sin(x)\) is \(- 1\leqslant\sin(x)\leqslant1\).

Step2: Find the range of \(y = 3\sin(x)\)

Multiply each part of the inequality \(-1\leqslant\sin(x)\leqslant1\) by \(3\). Using the property of inequalities, if \(a\leqslant b\leqslant c\) and \(k>0\), then \(ka\leqslant kb\leqslant kc\). Here \(k = 3\), \(a=-1\), \(b = \sin(x)\), \(c = 1\). So, \(3\times(-1)\leqslant3\sin(x)\leqslant3\times1\), which simplifies to \(-3\leqslant3\sin(x)\leqslant3\).

Answer:

The function's domain is all real numbers and the range is \(-3\leqslant y\leqslant3\).