https://file.aiquickdraw.com/mathbot/file/1725453464982ij83nsez.jpg

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.

Find the x-intercepts and y-intercepts of the given rational functions:
1. f(x) = (x - 8) / (x + 2)
2. f(x) = (x^2 - 10x + 25) / (x + 5)
3. f(x) = (x^2 + 9) / (x^2 - 1)

Learn how to find the x-intercepts and y-intercepts of rational functions with detailed step-by-step solutions. This guide covers key algebraic concepts, including how to determine intercepts for rational functions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation