https://file.aiquickdraw.com/mathbot/file/172680950194754pdv2bm.jpg

The function in the image is given as:

\[
f(x) = \frac{6}{-2x + 1}
\]

This is a rational function. Let’s break down key aspects of the function:

1. **Domain**: The denominator cannot be zero, as division by zero is undefined. To find the domain, set the denominator equal to zero and solve for \(x\):
   \[
   -2x + 1 = 0 \implies x = \frac{1}{2}
   \]
   So, the function is undefined at \(x = \frac{1}{2}\). The domain of \(f(x)\) is all real numbers except \(x = \frac{1}{2}\), i.e., \( x \in (-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty) \).

2. **Vertical Asymptote**: A vertical asymptote occurs where the denominator is zero. As calculated above, there is a vertical asymptote at \(x = \frac{1}{2}\).

3. **Horizontal Asymptote**: To find the horizontal asymptote, consider the degree of the numerator and the denominator. Since both have degree 1 (degree of \(x\)), we look at the ratio of the leading coefficients. In this case, the leading coefficient of the denominator is \(-2\) and the numerator is 6, so the horizontal asymptote is:
   \[
   y = \frac{6}{-2} = -3
   \]

Would you like to explore the graph or any specific behavior of the function? Here are some related questions:

1. How can we compute the intercepts (x- and y-intercepts) of the function?
2. Can you explain why the function has a vertical asymptote at \(x = \frac{1}{2}\)?
3. How does the function behave near the vertical asymptote \(x = \frac{1}{2}\)?
4. What is the significance of the horizontal asymptote \(y = -3\)?
5. How would you find the inverse of the function \(f(x)\)?

**Tip:** For rational functions like this, vertical asymptotes indicate where the function tends towards infinity or negative infinity, while horizontal asymptotes describe the function's behavior as \(x\) approaches large positive or negative values.

What is the behavior of the function f(x) = 6 / (-2x + 1)?

Explore the behavior of the rational function f(x) = 6 / (-2x + 1). Learn about its domain, vertical and horizontal asymptotes, and key properties. Suitable for Grades 10-12 students learning about rational functions and asymptotes.

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