Let `A={xinR:x" is not a positive integer "}`define a function `f:AtoR" such that "f(x)=(2x)/(x-1)`. Then f is

Options

A. injective but not surjective
B. not injective
C. surjective but not injective
D. neither inhective nor surjective

Answer

To determine the nature of the function \( f: A \to \mathbb{R} \) defined by \( f(x) = \frac{2x}{x - 1} \) where \( A = \{ x \in \mathbb{R} : x \text{ is not a positive integer} \} \), we need to check if the function is injective and/or surjective.

### Step 1: Check for Injectivity

A function is injective if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).

Assume \( f(x_1) = f(x_2) \):
\[ \frac{2x_1}{x_1 - 1} = \frac{2x_2}{x_2 - 1} \]

Cross-multiplying gives:
\[ 2x_1(x_2 - 1) = 2x_2(x_1 - 1) \]

Expanding both sides:
\[ 2x_1x_2 - 2x_1 = 2x_2x_1 - 2x_2 \]

Rearranging terms:
\[ -2x_1 = -2x_2 \]

Dividing both sides by -2:
\[ x_1 = x_2 \]

Thus, the function is injective.

### Step 2: Check for Surjectivity

A function is surjective if for every \( y \in \mathbb{R} \), there exists an \( x \in A \) such that \( f(x) = y \).

We start with:
\[ y = \frac{2x}{x - 1} \]

Rearranging gives:
\[ y(x - 1) = 2x \]
\[ yx - y = 2x \]
\[ yx - 2x = y \]
\[ x(y - 2) = y \]
\[ x = \frac{y}{y - 2} \]

Now we need to check if \( x = \frac{y}{y - 2} \) is in the domain \( A \) (i.e., \( x \) is not a positive integer).

1. If \( y = 2 \), then \( x \) is undefined.
2. For \( y \neq 2 \), we need to check if \( \frac{y}{y - 2} \) can be a positive integer.

Let \( \frac{y}{y - 2} = n \) (where \( n \) is a positive integer):
\[ y = n(y - 2) \]
\[ y = ny - 2n \]
\[ y - ny = -2n \]
\[ y(1 - n) = -2n \]
\[ y = \frac{-2n}{1 - n} \]

For \( n = 1 \), \( y = -2 \) (which is not a positive integer).
For \( n > 1 \), \( y \) becomes negative, and for \( n < 1 \), \( y \) becomes positive. Hence, there are values of \( y \) that do not yield a corresponding \( x \) in \( A \).

Thus, the function is not surjective.

### Conclusion

The function \( f(x) = \frac{2x}{x - 1} \) is **injective but not surjective**.