If f(x) =`(4x+x^(4))/(1+4x^(3))` and g(x) = In `(1+x)/(1-x)` t then what is the value of fog `(e-1)/(e+1)` equal to:

Options

A. 2
B. 1
C. 9
D. `1/2`

Answer

To find the value of \( f(g((e-1)/(e+1))) \), we will follow these steps:

### Step 1: Calculate \( g\left(\frac{e-1}{e+1}\right) \)

The function \( g(x) \) is defined as:
\[ g(x) = \ln\left(\frac{1+x}{1-x}\right) \]

Substituting \( x = \frac{e-1}{e+1} \):
\[ g\left(\frac{e-1}{e+1}\right) = \ln\left(\frac{1 + \frac{e-1}{e+1}}{1 - \frac{e-1}{e+1}}\right) \]

### Step 2: Simplify the expression inside the logarithm

Calculating the numerator:
\[ 1 + \frac{e-1}{e+1} = \frac{(e+1) + (e-1)}{e+1} = \frac{2e}{e+1} \]

Calculating the denominator:
\[ 1 - \frac{e-1}{e+1} = \frac{(e+1) - (e-1)}{e+1} = \frac{2}{e+1} \]

Now substituting back into \( g(x) \):
\[ g\left(\frac{e-1}{e+1}\right) = \ln\left(\frac{\frac{2e}{e+1}}{\frac{2}{e+1}}\right) = \ln(e) = 1 \]

### Step 3: Calculate \( f(g((e-1)/(e+1))) = f(1) \)

Now we need to find \( f(1) \) using the function \( f(x) \):
\[ f(x) = \frac{4x + x^4}{1 + 4x^3} \]

Substituting \( x = 1 \):
\[ f(1) = \frac{4(1) + (1)^4}{1 + 4(1)^3} = \frac{4 + 1}{1 + 4} = \frac{5}{5} = 1 \]

### Final Answer
Thus, the value of \( f(g((e-1)/(e+1))) \) is:
\[ \boxed{1} \]