If `f (x) =x ^(2) + 1/x, x ne 0` then `f ((1)/(x)) =`

Options

A. `(1)/(x ^(2)) + x`
B. ` 1/x + x^(2)`
C. `(1)/(x ^(2)) -x`
D. `1/x -x ^(2)`

Answer

To solve the problem, we need to find the value of \( f\left(\frac{1}{x}\right) \) given that \( f(x) = x^2 + \frac{1}{x} \) where \( x \neq 0 \).

### Step-by-Step Solution:

1. **Identify the function**: We have the function defined as:
\[ f(x) = x^2 + \frac{1}{x} \]

2. **Substitute \( \frac{1}{x} \) into the function**: We need to find \( f\left(\frac{1}{x}\right) \). To do this, we replace \( x \) in the function with \( \frac{1}{x} \):
\[ f\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^2 + \frac{1}{\frac{1}{x}} \]

3. **Calculate \( \left(\frac{1}{x}\right)^2 \)**: This simplifies to:
\[ \left(\frac{1}{x}\right)^2 = \frac{1}{x^2} \]

4. **Calculate \( \frac{1}{\frac{1}{x}} \)**: This simplifies to:
\[ \frac{1}{\frac{1}{x}} = x \]

5. **Combine the results**: Now we can combine both parts:
\[ f\left(\frac{1}{x}\right) = \frac{1}{x^2} + x \]

6. **Final expression**: Therefore, the final expression for \( f\left(\frac{1}{x}\right) \) is:
\[ f\left(\frac{1}{x}\right) = \frac{1}{x^2} + x \]

### Final Answer:
\[ f\left(\frac{1}{x}\right) = \frac{1}{x^2} + x \]