Question

$f\left(x\right)=4\tan x$

(a) Solve $f\left(x\right)=-4$. '

(b) For what values of x is $f\left(x\right)<-4$on the interval$\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$.

Short answer

(a) The solution is $\left\{x|x=-\frac{\mathrm{\pi }}{4}+n\mathrm{\pi },\mathrm{n}\mathrm{is}\mathrm{an}\mathrm{integer}\right\}$.

(b) The solution set is $\left\{-\frac{\mathrm{\pi }}{2},-\frac{\mathrm{\pi }}{4}\right\}$.

Step-by-step solution

Step 1. Given Information

$f\left(x\right)=4\tan x$

(a) Solve $f\left(x\right)=-4$. '

(b) For what values of x is$f\left(x\right)<-4$ on the interval$\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$.

Part (a) Step 1. The given equation is f(x)=4tanx

Putting $f\left(x\right)=-4$

$-4=4\tan x$

Divide by 4 on both side

$$\begin{gathered} -\frac{4}{4}=\frac{4}{4}\tan x \\ -1=\tan x \\ \tan x=-1 \end{gathered}$$

Part (a) Step 2. The solution for f(x)=-4. 

$\theta =-\frac{\mathrm{\pi }}{4}$

The solution is $\left\{x|x=-\frac{\mathrm{\pi }}{4}+n\mathrm{\pi },\mathrm{n}\mathrm{is}\mathrm{an}\mathrm{integer}\right\}$.

Step (b) Part 1. The values of x is f(x)<-4 on the interval -π2,π2.

$\theta =-\frac{\mathrm{\pi }}{2},-\frac{\mathrm{\pi }}{4}$

The solution set is$\left\{-\frac{\mathrm{\pi }}{2},-\frac{\mathrm{\pi }}{4}\right\}$