The multiplicative inverse of `(x+1)+(1)/(x-1)` is

Options

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

Explanation

To find the multiplicative inverse of the expression \((x + 1) + \frac{1}{(x - 1)}\), we will follow these steps:

### Step 1: Combine the terms
We start by simplifying the expression \((x + 1) + \frac{1}{(x - 1)}\). To do this, we need a common denominator, which is \((x - 1)\).

\[ (x + 1) + \frac{1}{(x - 1)} = \frac{(x + 1)(x - 1)}{(x - 1)} + \frac{1}{(x - 1)} \]

### Step 2: Simplify the expression
Now, we can combine the two fractions:

\[ = \frac{(x + 1)(x - 1) + 1}{(x - 1)} \]

Using the difference of squares, \((x + 1)(x - 1) = x^2 - 1\):

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

### Step 3: Find the multiplicative inverse
The multiplicative inverse of a number \(a\) is given by \(\frac{1}{a}\). Therefore, the multiplicative inverse of \(\frac{x^2}{(x - 1)}\) is:

\[ \text{Multiplicative Inverse} = \frac{(x - 1)}{x^2} \]

### Final Answer
Thus, the multiplicative inverse of \((x + 1) + \frac{1}{(x - 1)}\) is:

\[ \frac{x - 1}{x^2} \]

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