Let's analyze and solve the provided questions one by one.

1. Consider the function $f(x) = \frac{x+3}{x-1}$. It holds $f(x) \geq 0$ in:

   • To find where the function $f(x) \geq 0$, we need to determine where the numerator and denominator have the same sign. $$f(x) = \frac{x+3}{x-1}$$
     • $x + 3 \geq 0$ implies $x \geq -3$
     • $x - 1 > 0$ implies $x > 1$
     • $x - 1 < 0$ implies $x < 1$

   Combining these:

   • $\frac{x+3}{x-1} \geq 0$ in $(-\infty, -3] \cup [1, \infty)$

   Therefore, the correct answer is $(-\infty, -3] \cup [1, \infty)$.

2. The horizontal/oblique asymptote(s) of the function $f(x) = \frac{x+3}{x-1}$ is/are:

   • For horizontal asymptotes, since the degrees of the numerator and denominator are equal, we take the ratio of the leading coefficients. $$\lim_{x \to \infty} \frac{x+3}{x-1} = \lim_{x \to \infty} \frac{1 + \frac{3}{x}}{1 - \frac{1}{x}} = 1$$

   Therefore, the horizontal asymptote is $y = 1$.

3. The vertical asymptote(s) of the function $f(x) = \frac{x+3}{x-1}$ is/are:

   • Vertical asymptotes occur where the denominator is zero. $$x - 1 = 0 \implies x = 1$$

   Therefore, the vertical asymptote is $x = 1$.

4. The function $f(x) = \frac{x+3}{x-1}$ has the following extreme points:

   • To find extreme points, we first find the derivative $f'(x)$. $$f'(x) = \frac{(x-1) - (x+3)}{(x-1)^2} = \frac{-4}{(x-1)^2}$$
   • Setting $f'(x) = 0$ yields no solutions since the numerator is a constant and non-zero. Thus, we look at critical points and evaluate $f(x)$ at potential points given in the options. $$f(-1) = \frac{-1+3}{-1-1} = -1, \quad f(3) = \frac{3+3}{3-1} = 3, \quad f(1 \pm \sqrt{3})$$

   Therefore, the correct answer is $\text{maximum for } x = 1 - \sqrt{3}, \text{ minimum for } x = 1 + \sqrt{3}$.

5. The function $f(x) = \frac{x+3}{x-1}$ decreases in the interval:

   • Since the derivative $f'(x) = \frac{-4}{(x-1)^2}$ is always negative, $f(x)$ is always decreasing except at the vertical asymptote. Therefore, the correct answer is $(-\infty, 1) \cup (1, \infty)$.

Would you like further details or have any questions about these explanations?

Here are 5 more questions you can ask:

1. How do you find the derivative of a rational function?
2. What is the significance of horizontal asymptotes in a function?
3. How do you determine intervals of increase and decrease for a function?
4. Can you explain the process for finding vertical asymptotes?
5. How do extreme points relate to the derivative of a function?

Tip: When analyzing the behavior of a function, always check the signs of the numerator and denominator separately, especially for rational functions.