For `x in (0,(3)/(2)), " let " f(x)=sqrt(x),g(x) =tan x and h(x)=(1-x^(2))/(1+x^(2))`.
If `phi(x)=((hof)og)(x), " then " phi ((pi)/(3))` is equal to

Options

A. ` "tan"(pi)/(12)`
B. ` "tan"(11pi)/(12)`
C. ` "tan"(7pi)/(12)`
D. ` "tan"(5pi)/(12)`

Explanation

To solve the problem, we need to find the value of \( \phi\left(\frac{\pi}{3}\right) \) where \( \phi(x) = (h \circ f \circ g)(x) \). Let's break this down step by step.

### Step 1: Define the Functions
We have three functions defined as follows:
- \( f(x) = \sqrt{x} \)
- \( g(x) = \tan(x) \)
- \( h(x) = \frac{1 - x^2}{1 + x^2} \)

### Step 2: Find \( g\left(\frac{\pi}{3}\right) \)
First, we need to calculate \( g\left(\frac{\pi}{3}\right) \):
\[ g\left(\frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

### Step 3: Find \( f(g\left(\frac{\pi}{3}\right)) \)
Next, we find \( f(g\left(\frac{\pi}{3}\right)) = f(\sqrt{3}) \):
\[ f(\sqrt{3}) = \sqrt{\sqrt{3}} = 3^{1/4} = \sqrt[4]{3} \]

### Step 4: Find \( h(f(g\left(\frac{\pi}{3}\right))) \)
Now, we need to calculate \( h(f(g\left(\frac{\pi}{3}\right))) = h(\sqrt[4]{3}) \):
\[ h(\sqrt[4]{3}) = \frac{1 - (\sqrt[4]{3})^2}{1 + (\sqrt[4]{3})^2} = \frac{1 - \sqrt{3}}{1 + \sqrt{3}} \]

### Step 5: Simplify \( h(\sqrt[4]{3}) \)
To simplify \( \frac{1 - \sqrt{3}}{1 + \sqrt{3}} \), we can multiply the numerator and the denominator by the conjugate of the denominator:
\[ h(\sqrt[4]{3}) = \frac{(1 - \sqrt{3})(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} = \frac{(1 - \sqrt{3})^2}{1 - 3} = \frac{1 - 2\sqrt{3} + 3}{-2} = \frac{4 - 2\sqrt{3}}{-2} \]
\[ = -2 + \sqrt{3} \]

### Final Answer
Thus, the value of \( \phi\left(\frac{\pi}{3}\right) \) is:
\[ \phi\left(\frac{\pi}{3}\right) = -2 + \sqrt{3} \]