Question

the graph of the function \\(f(x) = \frac{6}{x-3}\\) is shown below.

what is the vertical asymptote of the function?

\\(x = -3\\)

Explanation:

Identify the rational function

The given function is:

$$f(x) = \frac{8}{x - 3}$$

Find the vertical asymptote algebraically

Set the denominator equal to zero to find where the function is undefined:

$$x - 3 = 0 \implies x = 3$$

Verify with the graph

Looking at the graph, as \(x\) approaches \(3\) from the left, \(f(x) \to -\infty\). As \(x\) approaches \(3\) from the right, \(f(x) \to \infty\). Thus, the vertical line \(x = 3\) is the vertical asymptote.

Answer:

• (A) \(x = -3\)
• (B) \(x = 3\) (Correct answer)
• (C) \(y = -3\)
• (D) \(y = 3\)