Question

which graph represents the function \\(f(x) = \frac{1}{x} - 1\\)?

Explanation:

Identify asymptotes and key features

The given function is:

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

• Vertical asymptote: \(x = 0\) (the y-axis).
• Horizontal asymptote: \(y = -1\).
• \(x\)-intercept: Set \(f(x) = 0 \implies \frac{1}{x} - 1 = 0 \implies x = 1\).
• Key points:
• For \(x = 1\), \(y = 0\).
• For \(x = -1\), \(y = -2\).

Analyze the given graph

• The visible graph has a vertical asymptote at \(x = 0\).
• The horizontal asymptote is at \(y = 1\).
• The \(x\)-intercept is at \(x = -1\).
• This graph represents \(y = \frac{1}{x} + 1\), which does not match \(f(x) = \frac{1}{x} - 1\).

Determine the correct graph behavior

• The correct graph of \(f(x) = \frac{1}{x} - 1\) must be shifted down by 1 unit compared to the parent function \(y = \frac{1}{x}\).
• Its horizontal asymptote must be \(y = -1\).
• It must pass through \((1, 0)\) and \((-1, -2)\).

Answer:

The correct graph representing the function \(f(x) = \frac{1}{x} - 1\) is the one that has a vertical asymptote at \(x = 0\), a horizontal asymptote at \(y = -1\), and passes through the points \((1, 0)\) and \((-1, -2)\).

Since the displayed graph has a horizontal asymptote at \(y = 1\) and passes through \((-1, 0)\), it represents \(y = \frac{1}{x} + 1\) and is incorrect. The correct option is the other graph choice (partially cut off at the bottom of the image) which is shifted downwards.

• (A) The displayed graph with horizontal asymptote at \(y = 1\)
• (B) The correct graph with horizontal asymptote at \(y = -1\) (Correct answer)