function tangent function: (f(x)=\tan x) sketch of graph write using interval notation intervals of…

function tangent function: (f(x)=\tan 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:

function tangent function: (f(x)=\tan 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:

Answer

Explanation:

Step1: Recall tangent function properties

The tangent function $y = \tan x=\frac{\sin x}{\cos x}$.

Step2: Determine domain

Since $\cos x = 0$ at $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$, the domain is $x\neq\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$, written in interval - notation as $\bigcup_{k\in\mathbb{Z}}\left(-\frac{\pi}{2}+k\pi,\frac{\pi}{2}+k\pi\right)$.

Step3: Determine range

The range of $y = \tan x$ is $(-\infty,\infty)$.

Step4: Find intervals of increasing and decreasing

The function $y=\tan x$ is increasing on intervals $\left(-\frac{\pi}{2}+k\pi,\frac{\pi}{2}+k\pi\right),k\in\mathbb{Z}$ and has no intervals of decreasing.

Step5: Find asymptotes

There are no horizontal asymptotes. The vertical asymptotes are at $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$.

Step6: Determine end - behavior

Left - end behavior: $\lim_{x\to\left(-\frac{\pi}{2}+k\pi\right)^+}\tan x=-\infty$; Right - end behavior: $\lim_{x\to\left(\frac{\pi}{2}+k\pi\right)^-}\tan x=\infty$.

Answer:

  • Sketch of Graph: The graph of $y = \tan x$ has vertical asymptotes at $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$, passes through the origin $(0,0)$ and is increasing on each of its intervals of continuity.
  • Intervals of Increasing: $\bigcup_{k\in\mathbb{Z}}\left(-\frac{\pi}{2}+k\pi,\frac{\pi}{2}+k\pi\right)$
  • Intervals of Decreasing: None
  • Left End Behavior: $\lim_{x\to\left(-\frac{\pi}{2}+k\pi\right)^+}\tan x=-\infty$
  • Right End Behavior: $\lim_{x\to\left(\frac{\pi}{2}+k\pi\right)^-}\tan x=\infty$
  • Domain: $\bigcup_{k\in\mathbb{Z}}\left(-\frac{\pi}{2}+k\pi,\frac{\pi}{2}+k\pi\right)$
  • Horizontal Asymptote(s): None
  • Vertical Asymptote(s): $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$
  • Range: $(-\infty,\infty)$