the function $g$ is given by $g(x)=\tan x$. what are all solutions to $g(x)=3$?\n(a) $x = arctan 3$ and…

the function $g$ is given by $g(x)=\tan x$. what are all solutions to $g(x)=3$?\n(a) $x = arctan 3$ and $x=pi+arctan 3$ only\n(b) $x = arctan 3$ and $x=pi+arctan(3 + pi)$ only\n(c) $x = 2pi k+arctan 3$ only, where $k$ is any integer\n(d) $x=pi k+arctan 3$, where $k$ is any integer

the function $g$ is given by $g(x)=\tan x$. what are all solutions to $g(x)=3$?\n(a) $x = arctan 3$ and $x=pi+arctan 3$ only\n(b) $x = arctan 3$ and $x=pi+arctan(3 + pi)$ only\n(c) $x = 2pi k+arctan 3$ only, where $k$ is any integer\n(d) $x=pi k+arctan 3$, where $k$ is any integer

Answer

Explanation:

Step1: Recall the inverse - tangent property

We know that if $\tan x = y$, then $x=\arctan y + n\pi$, where $n\in\mathbb{Z}$ (the set of all integers). The function $y = \tan x$ has a period of $\pi$. Given $\tan x=3$, the principal - value solution is $x = \arctan 3$.

Step2: Find all solutions

Since the period of the tangent function $y=\tan x$ is $\pi$, all solutions of the equation $\tan x = 3$ are given by $x=\pi k+\arctan 3$, where $k$ is an integer.

Answer:

D. $x = \pi k+\arctan 3$, where $k$ is any integer