The function `f(x)=(2x-1)/(x-2)(xne2)` is such that

Options

A. it is inverse of itself
B. decreases in `(-infty,2)` and `(2,infty)`
C. it has a graph entirely above x-axis
D. `underset(xto0^(+))(lim)f(e^(x^(-1))=2`

Explanation

To solve the problem, we need to analyze the function \( f(x) = \frac{2x - 1}{x - 2} \) for the given options. Let's go through each option step by step.

### Step 1: Finding the Inverse of the Function
1. Start with the function:
\[ y = \frac{2x - 1}{x - 2} \]
2. Rearranging to find \( x \) in terms of \( y \):
\[ y(x - 2) = 2x - 1 \]
\[ yx - 2y = 2x - 1 \]
3. Move all terms involving \( x \) to one side:
\[ yx - 2x = 2y - 1 \]
\[ x(y - 2) = 2y - 1 \]
4. Solve for \( x \):
\[ x = \frac{2y - 1}{y - 2} \]
5. Thus, the inverse function is:
\[ f^{-1}(y) = \frac{2y - 1}{y - 2} \]