Question

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

Explanation:

Step1: Analyze the parent function

The parent function is \( y = \frac{1}{x} \), which has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).

Step2: Analyze the transformation

For the function \( f(x)=\frac{1}{x}-1 \), it is a vertical translation of the parent function \( y = \frac{1}{x} \) down by 1 unit.
The vertical asymptote remains \( x = 0 \), and the horizontal asymptote becomes \( y=-1 \).

Answer:

The graph with vertical asymptote \( x = 0 \) and horizontal asymptote \( y=-1 \) (the one where the two branches of the hyperbola are shifted down by 1 unit from the basic \( y=\frac{1}{x} \) graph) is the correct representation.