part c: function $k$ is defined as $k(x)=\frac{sqrt{3}}{3}\tan(x + \frac{pi}{4})$. find all values in the…

Explanation:

Step1: Set up the equation

Set $k(x)= - 1$, so $\frac{\sqrt{3}}{3}\tan(x + \frac{\pi}{4})=-1$.

Step2: Solve for $\tan(x+\frac{\pi}{4})$

Divide both sides of the equation by $\frac{\sqrt{3}}{3}$, we get $\tan(x+\frac{\pi}{4})=- \sqrt{3}$.

Step3: Use the inverse - tangent property

We know that if $\tan\theta=- \sqrt{3}$, then $\theta = \frac{2\pi}{3}+n\pi$, where $n\in\mathbb{Z}$. So $x+\frac{\pi}{4}=\frac{2\pi}{3}+n\pi$.

Step4: Solve for $x$

Subtract $\frac{\pi}{4}$ from both sides: $x=\frac{2\pi}{3}-\frac{\pi}{4}+n\pi$. Find a common denominator: $x=\frac{8\pi - 3\pi}{12}+n\pi=\frac{5\pi}{12}+n\pi$, where $n\in\mathbb{Z}$.

Answer:

$x=\frac{5\pi}{12}+n\pi$, $n\in\mathbb{Z}$