Question

1. a tangent function of the form $y = \tan(bx)$ is defined for all values in the interval $(-2\pi, 2\pi)$. find a possible value of $b$.

Explanation:

Step1: Recall tangent domain rules

The function $y=\tan(u)$ is undefined when $u=\frac{\pi}{2}+k\pi$, where $k$ is any integer. For $u=bx$, the undefined points are $x=\frac{\pi}{2b}+\frac{k\pi}{b}$.

Step2: Set bounds for undefined points

We need all undefined points of $y=\tan(bx)$ to lie outside $(-2\pi, 2\pi)$. The closest undefined points to 0 are $x=\pm\frac{\pi}{2b}$. These must satisfy $\frac{\pi}{2b}\geq2\pi$.

Step3: Solve for $b$

$$\frac{\pi}{2b}\geq2\pi$$
Divide both sides by $\pi$:
$$\frac{1}{2b}\geq2$$
Rearrange to solve for $b$ (assuming $b>0$, as $b<0$ gives equivalent domain behavior):
$$1\geq4b$$
$$b\leq\frac{1}{4}$$
A positive value less than or equal to $\frac{1}{4}$ works.

Answer:

$\frac{1}{4}$ (any value $0<|b|\leq\frac{1}{4}$ is valid, $\frac{1}{4}$ is one example)