https://file.aiquickdraw.com/mathbot/file/1725891681133h2tzsujw.jpg

You need help with constructing a table of values and sketching the graph for the rational function \( f(x) = \frac{1}{x + 2} \).

### Step 1: Construct a Table of Values
We choose several values of \( x \), substitute them into the function, and compute \( f(x) \). Here’s a table for some common values:

| \( x \)   | \( f(x) = \frac{1}{x+2} \) |
|----------|---------------------------|
| -5       | \( f(-5) = \frac{1}{-5 + 2} = -\frac{1}{3} \)  |
| -4       | \( f(-4) = \frac{1}{-4 + 2} = -\frac{1}{2} \)  |
| -3       | \( f(-3) = \frac{1}{-3 + 2} = -1 \)            |
| -2       | \( f(-2) \) is undefined (as the denominator is 0)  |
| -1       | \( f(-1) = \frac{1}{-1 + 2} = 1 \)             |
| 0        | \( f(0) = \frac{1}{0 + 2} = \frac{1}{2} \)     |
| 1        | \( f(1) = \frac{1}{1 + 2} = \frac{1}{3} \)     |

### Step 2: Sketch the Graph
- **Asymptote**: The function has a vertical asymptote at \( x = -2 \) because the denominator becomes zero there.
- **Horizontal Asymptote**: As \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to 0 \). Hence, the horizontal asymptote is at \( y = 0 \).

The graph approaches these asymptotes but never touches them. It’s a standard hyperbola.

Do you have any questions or would you like more details on graphing?

### Related Questions:
1. How do you determine vertical and horizontal asymptotes for rational functions?
2. What happens when you shift the function \( \frac{1}{x} \) horizontally or vertically?
3. How would the graph of \( \frac{1}{x+2} + 1 \) differ from the current function?
4. Can the function \( f(x) = \frac{1}{x+2} \) ever touch the horizontal asymptote?
5. How do you solve rational equations that involve asymptotes?

**Tip**: When graphing rational functions, always check for undefined points (where the denominator is zero) and asymptotes for more accurate sketches.

Construct a table of values & sketch the graph of the following rational functions. f(x) = 1 / (x + 2)

Learn how to construct a table of values and graph the rational function f(x) = 1 / (x + 2). This solution involves key algebraic concepts, including determining asymptotes and calculating values of the function. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation