Question

function sine function: f(x)=sin x sketch of graph write using interval notation intervals of increasing intervals of decreasing left end behavior (use limit notation): right end behavior (use limit notation) write using interval notation domain horizontal asymptote(s): (write in the form of y = _ ) vertical asymptote(s). (write in the form of x = _ ) write using interval notation range

Explanation:

Step1: Recall properties of sine - function

The sine function \(y = \sin x\) is a periodic function with period \(2\pi\).

Step2: Find intervals of increasing

The derivative of \(y=\sin x\) is \(y'=\cos x\). The function is increasing when \(y'> 0\), i.e., \(\cos x>0\). The intervals of increasing are \((2k\pi-\frac{\pi}{2},2k\pi + \frac{\pi}{2})\), \(k\in\mathbb{Z}\).

Step3: Find intervals of decreasing

The function is decreasing when \(y'<0\), i.e., \(\cos x < 0\). The intervals of decreasing are \((2k\pi+\frac{\pi}{2},2k\pi+\frac{3\pi}{2})\), \(k\in\mathbb{Z}\).

Step4: Determine end - behavior

\(\lim_{x
ightarrow-\infty}\sin x\) and \(\lim_{x
ightarrow+\infty}\sin x\) do not exist. But the function oscillates between - 1 and 1.

Step5: Find domain

The domain of \(y = \sin x\) is all real numbers, which in interval notation is \((-\infty,\infty)\).

Step6: Find asymptotes

There are no vertical asymptotes since \(\sin x\) is defined for all real \(x\). There are no horizontal asymptotes since the function oscillates between - 1 and 1.

Step7: Find range

The range of \(y=\sin x\) is \([-1,1]\) in interval notation.

Answer:

• Intervals of increasing: \((2k\pi-\frac{\pi}{2},2k\pi+\frac{\pi}{2})\), \(k\in\mathbb{Z}\)
• Intervals of decreasing: \((2k\pi+\frac{\pi}{2},2k\pi+\frac{3\pi}{2})\), \(k\in\mathbb{Z}\)
• Left - end behavior: \(\lim_{x

ightarrow-\infty}\sin x\) does not exist (oscillates between - 1 and 1)

• Right - end behavior: \(\lim_{x

ightarrow+\infty}\sin x\) does not exist (oscillates between - 1 and 1)

• Domain: \((-\infty,\infty)\)
• Horizontal asymptote(s): None
• Vertical asymptote(s): None
• Range: \([-1,1]\)