Question

Question was provided via image upload.

Explanation:

Identify the given function

The function shown in the image is:

$$ f(x) = \frac{x}{x - 2} $$

This is a rational function defined for all \(x
eq 2\).

Find the inverse function

Using the Inverse Functions knowledge point:

$$
LATEXBLOCK0
$$

Interchanging \(x\) and \(y\), we get:

$$ f^{-1}(x) = \frac{2x}{x - 1} $$

Determine the domain of the inverse function

Using the Inverse Functions knowledge point:
The domain of \(f^{-1}(x)\) is the range of \(f(x)\).
For \(f^{-1}(x) = \frac{2x}{x - 1}\), the denominator cannot be zero:

$$ x - 1 eq 0 \implies x eq 1 $$

Thus, the domain is \(x
eq 1\).

Select the correct option

The options are:

• A. \(f^{-1}(x) = \Box\), for \(x \le \Box\)
• B. \(f^{-1}(x) = \Box\), for \(x \ge \Box\)
• C. \(f^{-1}(x) = \Box\), for \(x

eq \Box\)

• D. \(f^{-1}(x) = \Box\), for all \(x\)

Option C matches our result:

$$ f^{-1}(x) = \frac{2x}{x - 1}, \quad \text{for } x eq 1 $$

Answer:

• (A) \(f^{-1}(x) = \frac{2x}{x - 1}\), for \(x \le 1\)
• (B) \(f^{-1}(x) = \frac{2x}{x - 1}\), for \(x \ge 1\)
• **(C) \(f^{-1}(x) = \frac{2x}{x - 1}\), for \(x

eq 1\) (Correct answer)**

• (D) \(f^{-1}(x) = \frac{2x}{x - 1}\), for all \(x\)