your turn\n1. write the function rule for the transformed tangent function of the form f(x)=a tan(1/b x)…

Explanation:

Step1: Substitute the point into the function

We know the function is (f(x)=a\tan(\frac{1}{b}x)) and the point ((\frac{1}{8},\frac{1}{2})) lies on the graph. So we substitute (x = \frac{1}{8}) and (y=f(x)=\frac{1}{2}) into the function: (\frac{1}{2}=a\tan(\frac{1}{b}\times\frac{1}{8})).

Step2: Consider the period - related property of the tangent function

The standard period of (y = \tan(x)) is (\pi). For the function (y=\tan(\frac{1}{b}x)), the period (T = b\pi). From the graph, we can observe that the period (T = 1), so (b\pi=1), then (b=\frac{1}{\pi}).

Step3: Find the value of (a)

Substitute (b = \frac{1}{\pi}) into (\frac{1}{2}=a\tan(\frac{1}{b}\times\frac{1}{8})), we get (\frac{1}{2}=a\tan(\frac{\pi}{1}\times\frac{1}{8})=a\tan(\frac{\pi}{8})). Since (\tan(\frac{\pi}{8})=\sqrt{2}- 1), then (a=\frac{1}{2(\sqrt{2}-1)}=\frac{\sqrt{2}+1}{2}).

Answer:

(f(x)=\frac{\sqrt{2}+1}{2}\tan(\pi x))