For the given rational function

f left parenthesis x right parenthesis equals StartFraction 6 minus 3 x Over x minus 6 EndFractionf(x)=6−3xx−6​,

find the following.

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.

Explore the analysis of the rational function f(x) = (6 - 3x) / (x - 6). Learn about its domain, vertical and horizontal asymptotes, x-intercept, and y-intercept with detailed explanations and examples. Suitable for Grades 10-12.

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