which of the following is the graph of this rational function? y = 1/x - 1

Explanation:

Step1: Analyze vertical asymptote

The function $y = \frac{1}{x}-1$ has a vertical asymptote at $x = 0$ since the denominator of the rational - part $\frac{1}{x}$ cannot be zero.

Step2: Analyze horizontal asymptote

As $x\to\pm\infty$, $\frac{1}{x}\to0$, so $y=\frac{1}{x}-1\to - 1$. The horizontal asymptote is $y=-1$.

Step3: Analyze y - intercept

Set $x = 0$, then $y=\frac{1}{0}-1$ is undefined. To find the y - intercept, set $x = 1$, $y=\frac{1}{1}-1=0$.

Step4: Analyze x - intercept

Set $y = 0$, then $\frac{1}{x}-1 = 0$, $\frac{1}{x}=1$, and $x = 1$.

The graph of $y=\frac{1}{x}-1$ is a hyperbola that has been shifted down 1 unit from the graph of $y=\frac{1}{x}$. It has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y=-1$, and passes through the point $(1,0)$.

Answer:

Without seeing the specific details of the graphs, the correct graph should have a vertical asymptote at $x = 0$, a horizontal asymptote at $y=-1$, and pass through the point $(1,0)$. If you can provide more details about the graphs (such as key - points on them), a more definite answer can be given.