The reciprocal of `x + 1/x` is

Options

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

Answer

To find the reciprocal of \( x + \frac{1}{x} \), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ x + \frac{1}{x} \]

### Step 2: Combine the terms
To combine the terms into a single fraction, we can find a common denominator. The common denominator here is \( x \):
\[ x + \frac{1}{x} = \frac{x^2}{x} + \frac{1}{x} = \frac{x^2 + 1}{x} \]

### Step 3: Find the reciprocal
The reciprocal of a fraction \( \frac{a}{b} \) is given by \( \frac{b}{a} \). Therefore, the reciprocal of \( \frac{x^2 + 1}{x} \) is:
\[ \frac{x}{x^2 + 1} \]

### Final Answer
Thus, the reciprocal of \( x + \frac{1}{x} \) is:
\[ \frac{x}{x^2 + 1} \]

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