If f(x)= tan(x) and g(x)= 3𝑥2x 2 , then (𝑔∘𝑓)(𝜋6)=(g∘f)( 6π​ )=A.1B.3(𝜋6)23( 6π​ ) 2 C.tan⁡((3𝜋6)2)tan(( 63π​ ) 2 )closeD.1.077E.None of the above

1. Break Down the Problem

We need to find the value of $(g \circ f)(\frac{\pi}{6})$ and $(g \circ f)(6\pi)$. This requires calculating $f(x)$ first and then plugging the result into $g(x)$.

2. Relevant Concepts

1. The function $f(x) = \tan(x)$.
2. The function $g(x) = 3x^2$.
3. The composition of functions $(g \circ f)(x) = g(f(x))$.

3. Analysis and Detail

1. Calculate $f\left(\frac{\pi}{6}\right)$: $f\left(\frac{\pi}{6}\right) = \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$

2. Now, calculate $g\left(f\left(\frac{\pi}{6}\right)\right)$: $g\left(f\left(\frac{\pi}{6}\right)\right) = g\left(\frac{1}{\sqrt{3}}\right) = 3\left(\frac{1}{\sqrt{3}}\right)^2 = 3 \cdot \frac{1}{3} = 1$

3. Next, calculate $f(6\pi)$: $f(6\pi) = \tan(6\pi) = 0$

4. Now, calculate $g\left(f(6\pi)\right)$: $g\left(f(6\pi)\right) = g(0) = 3(0)^2 = 0$

4. Verify and Summarize

• From our calculations:
  • $(g \circ f)\left(\frac{\pi}{6}\right) = 1$
  • $(g \circ f)(6\pi) = 0$

Final Answer

Since the question asks for the values:

• $(g \circ f)(\frac{\pi}{6}) = 1$
• $(g \circ f)(6\pi) = 0$

The answer corresponding to $(g \circ f)(\frac{\pi}{6})$ is A. 1.