A function with a horizontal asymptote at $y = 3$ and vertical asymptotes at $x = 0$ and $x = 2$ could be a rational function of the form:

$f(x) = \frac{3(x - a)}{(x)(x - 2)}$

Explanation:

1. Horizontal Asymptote at $y = 3$:

   • The horizontal asymptote occurs when the degree of the numerator is equal to the degree of the denominator. The ratio of the leading coefficients of the numerator and denominator determines the horizontal asymptote. In this case, the horizontal asymptote is $y = 3$, so the leading coefficient of the numerator must be 3.

2. Vertical Asymptotes at $x = 0$ and $x = 2$:

   • Vertical asymptotes occur where the denominator is zero, so we need $x = 0$ and $x = 2$ to be the roots of the denominator. Thus, the denominator must include $(x)(x - 2)$.

Example function:

A simple function that satisfies these conditions is:

$f(x) = \frac{3}{x(x - 2)}$

This function has:

• A horizontal asymptote at $y = 3$ as $x \to \infty$.
• Vertical asymptotes at $x = 0$ and $x = 2$.

Would you like further details or have any questions?

Here are 5 related questions:

1. What are the steps to determine horizontal and vertical asymptotes for rational functions?
2. How do you find the oblique (slant) asymptotes of a function?
3. Can you describe how the behavior of a function changes near its vertical asymptotes?
4. How can we adjust this function to shift the horizontal asymptote to a different value, like $y = 5$?
5. What happens to the graph of this function as $x \to -\infty$?

Tip: A horizontal asymptote shows the end behavior of a function as $x \to \infty$ or $x \to -\infty$.