Cole Ramirez
09/29/2024 · High School
I. Work Out (Show the necessary steps) For the rational functions (a-d) given below answer question (i-vi) each of the following i. Find the domain ii. \( \quad \) Find hole, if any iii. Find the vertical asymptote(s), if any iv. Find the horizontal or oblique asymptote, if any v. Find the \( x \)-intercept(s) and \( y \) - intercept(s) vi. \( \quad \) Prepare sign chart vii. Sketch the graph of each rational function a. \( \quad f(x)=\frac{x+2}{x^{2}-9} \)
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**Analysis of \( f(x) = \frac{x+2}{x^2 - 9} \):**
- **Domain:** All real numbers except \( x = 3 \) and \( x = -3 \).
- **Holes:** None.
- **Vertical Asymptotes:** \( x = 3 \) and \( x = -3 \).
- **Horizontal Asymptote:** \( y = 0 \).
- **Intercepts:** \( x \)-intercept at \( (-2, 0) \) and \( y \)-intercept at \( (0, -\frac{2}{9}) \).
- **Sign Chart:**
- \( (-\infty, -3) \): Negative
- \( (-3, -2) \): Positive
- \( (-2, 3) \): Negative
- \( (3, \infty) \): Positive
- **Graph:** Approaches vertical asymptotes at \( x = 3 \) and \( x = -3 \), crosses the \( x \)-axis at \( (-2, 0) \), and the \( y \)-axis at \( (0, -\frac{2}{9}) \), with the horizontal asymptote at \( y = 0 \).
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