Let's analyze the image you uploaded, which contains three rational functions. The task is to find the $x$-intercepts and $y$-intercepts for each function.

Function 1: $f(x) = \frac{x - 8}{x + 2}$

• $x$-intercept: Set $f(x) = 0$: $\frac{x - 8}{x + 2} = 0$ This implies that the numerator must be zero: $x - 8 = 0 \quad \Rightarrow \quad x = 8$ So, the $x$-intercept is $(8, 0)$.

• $y$-intercept: Set $x = 0$: $f(0) = \frac{0 - 8}{0 + 2} = \frac{-8}{2} = -4$ So, the $y$-intercept is $(0, -4)$.

Function 2: $f(x) = \frac{x^2 - 10x + 25}{x + 5}$

• $x$-intercept: Set $f(x) = 0$: $\frac{x^2 - 10x + 25}{x + 5} = 0$ The numerator must be zero: $x^2 - 10x + 25 = (x - 5)^2 = 0 \quad \Rightarrow \quad x = 5$ So, the $x$-intercept is $(5, 0)$.

• $y$-intercept: Set $x = 0$: $f(0) = \frac{0^2 - 10(0) + 25}{0 + 5} = \frac{25}{5} = 5$ So, the $y$-intercept is $(0, 5)$.

Function 3: $f(x) = \frac{x^2 + 9}{x^2 - 1}$

• $x$-intercept: Set $f(x) = 0$: $\frac{x^2 + 9}{x^2 - 1} = 0$ Since $x^2 + 9$ is always positive for all real numbers $x$, it can never be zero. Therefore, there are no $x$-intercepts.

• $y$-intercept: Set $x = 0$: $f(0) = \frac{0^2 + 9}{0^2 - 1} = \frac{9}{-1} = -9$ So, the $y$-intercept is $(0, -9)$.

Summary:

1. $f(x) = \frac{x - 8}{x + 2}$:

   • $x$-intercept: $(8, 0)$
   • $y$-intercept: $(0, -4)$

2. $f(x) = \frac{x^2 - 10x + 25}{x + 5}$:

   • $x$-intercept: $(5, 0)$
   • $y$-intercept: $(0, 5)$

3. $f(x) = \frac{x^2 + 9}{x^2 - 1}$:

   • $x$-intercept: None
   • $y$-intercept: $(0, -9)$

Would you like more details on any of these steps?

Here are some follow-up questions:

1. What happens to the function's graph near points where the denominator is zero?
2. How do we find vertical asymptotes of a rational function?
3. What is the behavior of $f(x)$ as $x$ approaches infinity for each function?
4. How would the intercepts change if the functions were altered slightly?
5. Can rational functions have slant asymptotes?

Tip: Remember, for rational functions, an $x$-intercept occurs when the numerator equals zero, and the $y$-intercept is found by substituting $x = 0$ into the function.