To determine the equation of the horizontal asymptote for the function $f(x) = \frac{x - 6}{2x + 1}$, follow these steps:

Step 1: Identify the degrees of the numerator and denominator.

• The numerator $x - 6$ has a degree of 1 (since $x$ is to the power of 1).
• The denominator $2x + 1$ also has a degree of 1.

Step 2: Compare the degrees of the numerator and denominator.

• When the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients.

Step 3: Find the leading coefficients.

• The leading coefficient of the numerator (the coefficient of $x$) is 1.
• The leading coefficient of the denominator is 2.

Step 4: Determine the horizontal asymptote.

• The horizontal asymptote is $y = \frac{1}{2}$.

So, the equation of the horizontal asymptote is $y = \frac{1}{2}$.

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. What is the vertical asymptote of the function $f(x) = \frac{x - 6}{2x + 1}$?
2. How would you find the x-intercept of the function $f(x) = \frac{x - 6}{2x + 1}$?
3. What happens to the horizontal asymptote if the degrees of the numerator and denominator are different?
4. Can a rational function have more than one horizontal asymptote? Why or why not?
5. How does the horizontal asymptote affect the behavior of the function as $x$ approaches infinity?

Tip: Always check the degrees of the numerator and denominator when analyzing horizontal asymptotes in rational functions.