Find the vertical asymptote of the function r(x) = 4x + 1/x - 3

To find the vertical asymptote(s) of the function \( r(x) = \frac{4x + 1}{x - 3} \), we follow these steps:

### Step 1: Identify the denominator.
A vertical asymptote occurs where the function is undefined, which happens when the denominator is equal to zero.

For the function \( r(x) = \frac{4x + 1}{x - 3} \), the denominator is \( x - 3 \).

### Step 2: Set the denominator equal to zero and solve.
\[
x - 3 = 0
\]
Solving for \( x \):
\[
x = 3
\]

### Conclusion:
The vertical asymptote of the function is at \( x = 3 \).

Would you like more details on this process or have any additional questions?

Here are 5 related questions:
1. What is the horizontal asymptote of this function?
2. How do you find the horizontal asymptote for rational functions in general?
3. What is the behavior of the function near the vertical asymptote?
4. Can this function have oblique (slant) asymptotes? Why or why not?
5. How do vertical asymptotes affect the graph of a rational function?

**Tip:** Always check both the numerator and the denominator for any common factors before finding vertical asymptotes. If they cancel out, there may not be a vertical asymptote at that point.

Learn how to find the vertical asymptote of the rational function r(x) = (4x + 1)/(x - 3). This step-by-step guide explains how vertical asymptotes occur when the denominator is zero. Perfect for high school students studying rational functions in Grades 9-12.

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