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:

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.