the graph of the function $y = \\tan(x)$ was horizontally stretched so that its period became $10\\pi$…

Explanation:

Step1: Recall period - formula for tangent function

The period of the tangent function $y = A\tan(Bx - C)+D$ is $T=\frac{\pi}{|B|}$. The original function $y = \tan(x)$ has a period of $\pi$ (since for $y=\tan(x)$, $B = 1$ and $T=\frac{\pi}{|1|}=\pi$).

Step2: Set up equation for new - period

We want the new period $T = 10\pi$. Using the formula $T=\frac{\pi}{|B|}$, and substituting $T = 10\pi$, we get $10\pi=\frac{\pi}{|B|}$.

Step3: Solve for B

Cross - multiply the equation $10\pi=\frac{\pi}{|B|}$ to get $10\pi|B|=\pi$. Then divide both sides by $10\pi$: $|B|=\frac{1}{10}$. Since we are dealing with a horizontal stretch, $B=\frac{1}{10}$. So the transformed function is $y = \tan(\frac{x}{10})$.

Answer:

A. $y=\tan(\frac{x}{10})$