If `f(x) = 7/(x^7) + 5/(x^5) + 3/(x^3) + 1 + 3x^3 + 5x^5 + 7x^7`. Now, if the value of `f(2)` is `1081.58` then the value of `f(1/2)` is:

Options

A. 540.79
B. `1/(1081.58)`
C. `1081.58`
D. `367.42`

Answer

To find the value of \( f\left(\frac{1}{2}\right) \) given that \( f(2) = 1081.58 \), we can use the property of the function \( f(x) \).

### Step-by-step Solution:

1. **Understanding the Function**: The function is given as:
\[ f(x) = \frac{7}{x^7} + \frac{5}{x^5} + \frac{3}{x^3} + 1 + 3x^3 + 5x^5 + 7x^7 \]

2. **Substituting \( x = 2 \)**: We know that \( f(2) = 1081.58 \).

3. **Finding \( f\left(\frac{1}{2}\right) \)**: We can substitute \( x = \frac{1}{2} \) into the function:
\[ f\left(\frac{1}{2}\right) = \frac{7}{\left(\frac{1}{2}\right)^7} + \frac{5}{\left(\frac{1}{2}\right)^5} + \frac{3}{\left(\frac{1}{2}\right)^3} + 1 + 3\left(\frac{1}{2}\right)^3 + 5\left(\frac{1}{2}\right)^5 + 7\left(\frac{1}{2}\right)^7 \]

4. **Simplifying Each Term**:
- \( \frac{7}{\left(\frac{1}{2}\right)^7} = 7 \cdot 2^7 = 7 \cdot 128 = 896 \)
- \( \frac{5}{\left(\frac{1}{2}\right)^5} = 5 \cdot 2^5 = 5 \cdot 32 = 160 \)
- \( \frac{3}{\left(\frac{1}{2}\right)^3} = 3 \cdot 2^3 = 3 \cdot 8 = 24 \)
- \( 1 = 1 \)
- \( 3\left(\frac{1}{2}\right)^3 = 3 \cdot \frac{1}{8} = \frac{3}{8} = 0.375 \)
- \( 5\left(\frac{1}{2}\right)^5 = 5 \cdot \frac{1}{32} = \frac{5}{32} = 0.15625 \)
- \( 7\left(\frac{1}{2}\right)^7 = 7 \cdot \frac{1}{128} = \frac{7}{128} \approx 0.0546875 \)

5. **Combining All Terms**:
\[ f\left(\frac{1}{2}\right) = 896 + 160 + 24 + 1 + 0.375 + 0.15625 + 0.0546875 \]
\[ = 1081.5859375 \]

6. **Conclusion**: Since \( f(2) = 1081.58 \) and the calculations show that \( f\left(\frac{1}{2}\right) \) is approximately equal to \( 1081.58 \), we conclude that:
\[ f\left(\frac{1}{2}\right) = 1081.58 \]

### Final Answer:
The value of \( f\left(\frac{1}{2}\right) \) is \( 1081.58 \).