From the image, the task is to graph the rational function $r(x) = \frac{2x - 3}{x - 1}$ using transformations of the basic graph of $y = \frac{1}{x}$. Additionally, you need to determine the domain and range of the function using interval notation.

Let's break this down:

1. Graph Selection:

You are presented with multiple graphs and you need to choose the correct one based on the transformation of $y = \frac{1}{x}$ to fit $r(x) = \frac{2x - 3}{x - 1}$.

The rational function $r(x) = \frac{2x - 3}{x - 1}$ involves:

• Vertical asymptote: This occurs where the denominator is zero. So, $x - 1 = 0$, giving a vertical asymptote at $x = 1$.
• Horizontal asymptote: The degrees of the numerator and denominator are both 1 (they're both linear). Thus, the horizontal asymptote is determined by the ratio of the leading coefficients. This gives $y = \frac{2}{1} = 2$.
• Shifts: The graph of $y = \frac{1}{x}$ is shifted right by 1 unit and vertically stretched by a factor of 2.

Therefore, the correct graph should show:

• A vertical asymptote at $x = 1$,
• A horizontal asymptote at $y = 2$,
• A curve that resembles $\frac{1}{x}$ but transformed as described above.

Based on this, the leftmost graph (which you selected) seems to represent the correct transformation, as it shows a vertical asymptote at $x = 1$ and a horizontal asymptote at $y = 2$.

2. Domain and Range:

• Domain: The domain of $r(x)$ excludes the value where the denominator is zero, i.e., $x = 1$. So, the domain is $(-\infty, 1) \cup (1, \infty)$.
• Range: The range excludes the value of the horizontal asymptote, i.e., $y = 2$. So, the range is $(-\infty, 2) \cup (2, \infty)$.

Final Answer:

• Domain: $(-\infty, 1) \cup (1, \infty)$
• Range: $(-\infty, 2) \cup (2, \infty)$

Would you like further details on how transformations of rational functions work?

Here are 5 related questions to deepen your understanding:

1. How can I use transformations to graph other rational functions?
2. How do I determine the asymptotes of a more complex rational function?
3. What is the significance of the degree of the numerator and denominator in rational functions?
4. How do vertical and horizontal shifts affect the graph of $y = \frac{1}{x}$?
5. Can rational functions have slant (oblique) asymptotes, and how do I find them?

Tip: Always look at the degree of the numerator and denominator to help find horizontal or slant asymptotes!