Question

function cosine function: (f(x)=cos 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 cosine - function properties

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

Step2: Determine intervals of increasing

The cosine function \(y=\cos x\) is increasing on the intervals \([2k\pi-\pi,2k\pi]\), \(k\in\mathbb{Z}\). In interval - notation, the set of all such intervals is \(\bigcup_{k = -\infty}^{\infty}[2k\pi-\pi,2k\pi]\).

Step3: Determine intervals of decreasing

The cosine function \(y = \cos x\) is decreasing on the intervals \([2k\pi,2k\pi+\pi]\), \(k\in\mathbb{Z}\). In interval - notation, the set of all such intervals is \(\bigcup_{k=-\infty}^{\infty}[2k\pi,2k\pi + \pi]\).

Step4: Find left - end behavior

\(\lim_{x
ightarrow-\infty}\cos x\) does not exist, but \(- 1\leqslant\cos x\leqslant1\) for all \(x\in\mathbb{R}\).

Step5: Find right - end behavior

\(\lim_{x
ightarrow\infty}\cos x\) does not exist, but \(-1\leqslant\cos x\leqslant1\) for all \(x\in\mathbb{R}\).

Step6: Determine domain

The domain of \(y = \cos x\) is all real numbers. In interval notation, the domain is \((-\infty,\infty)\).

Step7: Find horizontal asymptotes

Since \(\lim_{x
ightarrow\pm\infty}\cos x\) does not exist and \(-1\leqslant\cos x\leqslant1\) for all \(x\in\mathbb{R}\), there are no horizontal asymptotes.

Step8: Find vertical asymptotes

The cosine function \(y=\cos x\) is a continuous function for all real \(x\), so there are no vertical asymptotes.

Step9: Determine range

The range of \(y = \cos x\) is \([-1,1]\) because the maximum value of \(\cos x\) is \(1\) and the minimum value is \(-1\).

Answer:

• Intervals of increasing: \(\bigcup_{k = -\infty}^{\infty}[2k\pi-\pi,2k\pi]\)
• Intervals of decreasing: \(\bigcup_{k=-\infty}^{\infty}[2k\pi,2k\pi + \pi]\)
• Left - end behavior: \(\lim_{x

ightarrow-\infty}\cos x\) does not exist, \(-1\leqslant\cos x\leqslant1\)

• Right - end behavior: \(\lim_{x

ightarrow\infty}\cos x\) does not exist, \(-1\leqslant\cos x\leqslant1\)

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