Question

which of the following could be the function graphed?

\\(f(x) = \frac{8x^2}{x^2 - 9}\\)

\\(f(x) = \frac{119x}{792x + 345}\\)

\\(f(x) = \frac{612}{816x - 272}\\)

\\(f(x) = \frac{237x}{421x - 515}\\)

Explanation:

Identify key features from the graph

The graph shows a rational function with:

1. A vertical asymptote in the first quadrant, indicating a positive vertical asymptote \(x = c > 0\).
2. A horizontal asymptote \(y = d > 0\) in the first quadrant.
3. An \(x\)-intercept at the origin \((0,0)\), meaning \(f(0) = 0\).

Analyze the given options

1. \(f(x) = \frac{8x^2}{x^2-9}\):

• Vertical asymptotes at \(x = \pm 3\).
• Horizontal asymptote at \(y = 8\).
• This function has two vertical asymptotes, but the graph only shows one.

1. \(f(x) = \frac{119x}{792x+345}\):

• Vertical asymptote at \(792x + 345 = 0 \implies x = -\frac{345}{792} < 0\).
• The graph has a positive vertical asymptote, so this is incorrect.

1. \(f(x) = \frac{612}{816x-272}\):

• \(x\)-intercept: None (numerator is a constant).
• The graph passes through \((0,0)\), so this is incorrect.

1. \(f(x) = \frac{237x}{421x-515}\):

• Vertical asymptote at \(421x - 515 = 0 \implies x = \frac{515}{421} > 0\).
• Horizontal asymptote at \(y = \frac{237}{421} > 0\).
• \(x\)-intercept at \(x = 0\).
• This matches all visual features of the graph.

Confirm the correct option

The fourth option \(f(x) = \frac{237x}{421x-515}\) is the only function that matches all the graphical characteristics.

Answer:

• (A) \(f(x) = \frac{8x^2}{x^2-9}\)
• (B) \(f(x) = \frac{119x}{792x+345}\)
• (C) \(f(x) = \frac{612}{816x-272}\)
• (D) \(f(x) = \frac{237x}{421x-515}\) (Correct answer)