If f(x) = `(x^(2))/(1 + x^(2))` , then the value of f{f(2)} is :

Options

A. `(9)/(41)`
B. `(25)/(41)`
C. `(16)/(25)`
D. `(16)/(41)`

Answer

To find the value of \( f(f(2)) \) where \( f(x) = \frac{x^2}{1 + x^2} \), we will follow these steps:

### Step 1: Calculate \( f(2) \)
We start by substituting \( x = 2 \) into the function \( f(x) \):

\[ f(2) = \frac{2^2}{1 + 2^2} \]

Calculating \( 2^2 \):

\[ 2^2 = 4 \]

Now substituting back into the function:

\[ f(2) = \frac{4}{1 + 4} = \frac{4}{5} \]

### Step 2: Calculate \( f(f(2)) \)
Now we need to find \( f(f(2)) \), which is \( f\left(\frac{4}{5}\right) \). We substitute \( x = \frac{4}{5} \) into the function \( f(x) \):

\[ f\left(\frac{4}{5}\right) = \frac{\left(\frac{4}{5}\right)^2}{1 + \left(\frac{4}{5}\right)^2} \]

Calculating \( \left(\frac{4}{5}\right)^2 \):

\[ \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]

Now substituting this back into the function:

\[ f\left(\frac{4}{5}\right) = \frac{\frac{16}{25}}{1 + \frac{16}{25}} \]

### Step 3: Simplify the denominator
To simplify the denominator, we need to add \( 1 + \frac{16}{25} \):

\[ 1 = \frac{25}{25} \quad \text{so} \quad 1 + \frac{16}{25} = \frac{25 + 16}{25} = \frac{41}{25} \]

### Step 4: Substitute back into the function
Now we substitute this back into our expression for \( f\left(\frac{4}{5}\right) \):

\[ f\left(\frac{4}{5}\right) = \frac{\frac{16}{25}}{\frac{41}{25}} \]

### Step 5: Simplify the fraction
When dividing fractions, we multiply by the reciprocal:

\[ f\left(\frac{4}{5}\right) = \frac{16}{25} \times \frac{25}{41} = \frac{16}{41} \]

### Final Answer
Thus, the value of \( f(f(2)) \) is:

\[ \boxed{\frac{16}{41}} \]