To analyze the given rational function

$f(x) = \frac{6 - 3x}{x - 6},$

we'll find the following:

1. Domain of the function:

The domain of a rational function includes all real numbers except where the denominator is equal to zero.

Set the denominator equal to zero and solve for $x$: $x - 6 = 0 \implies x = 6.$ Therefore, the domain of $f(x)$ is all real numbers except $x = 6$: $\text{Domain: } (-\infty, 6) \cup (6, \infty).$

2. Vertical asymptote:

Vertical asymptotes occur where the denominator of the function equals zero (and the numerator does not equal zero at those points).

Since $x = 6$ makes the denominator zero, and the numerator $6 - 3x$ does not equal zero at $x = 6$ (it equals $-12$), there is a vertical asymptote at: $x = 6.$

3. Horizontal asymptote:

To find the horizontal asymptote, compare the degrees of the numerator and the denominator:

• Both the numerator and the denominator are linear (degree 1).
• The horizontal asymptote is given by the ratio of the leading coefficients of the numerator and denominator.

The leading coefficient of the numerator ($-3x$) is $-3$, and the leading coefficient of the denominator ($x$) is $1$. Thus, the horizontal asymptote is: $y = \frac{-3}{1} = -3.$

4. X-intercept:

The x-intercept(s) occur where the numerator equals zero. Set the numerator equal to zero and solve for $x$: $6 - 3x = 0 \implies x = 2.$ So, the x-intercept is at: $(2, 0).$

5. Y-intercept:

The y-intercept occurs where $x = 0$. Substitute $x = 0$ into the function: $f(0) = \frac{6 - 3(0)}{0 - 6} = \frac{6}{-6} = -1.$ So, the y-intercept is at: $(0, -1).$

Summary:

• Domain: $(-\infty, 6) \cup (6, \infty)$
• Vertical asymptote: $x = 6$
• Horizontal asymptote: $y = -3$
• X-intercept: $(2, 0)$
• Y-intercept: $(0, -1)$

Would you like any additional details or have any questions?

Here are five related questions:

1. How would you find the slant asymptote if the degrees of the numerator and denominator were different?
2. What happens to the graph near the vertical asymptote?
3. Can a rational function have more than one horizontal asymptote?
4. What does the x-intercept tell us about the function?
5. How would the function behave if the numerator and denominator had the same factors?

Tip: Always check for common factors between the numerator and denominator, as canceling them can reveal holes in the graph rather than vertical asymptotes.