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To find the **domain** and **range** of the given function:

\[
f(x) = \frac{1}{x + 2}
\]

### Domain:
The domain of a function is the set of all possible input values (x) that make the function well-defined. In this case, the function has a denominator, and we must ensure that the denominator is not zero.

The denominator \( x + 2 \) is zero when:
\[
x + 2 = 0 \implies x = -2
\]
Thus, the function is undefined when \( x = -2 \).

Therefore, the **domain** is all real numbers except \( x = -2 \), which can be written as:
\[
\text{Domain}: \{ x \in \mathbb{R} \mid x \neq -2 \}
\]
or in interval notation:
\[
\text{Domain}: (-\infty, -2) \cup (-2, +\infty)
\]

### Range:
The range of a function is the set of all possible output values (f(x)) for the domain.

For the given function \( f(x) = \frac{1}{x + 2} \), the output will never be zero because no value of \( x \) makes \( \frac{1}{x + 2} = 0 \).

Also, as \( x \) approaches \( -2 \), the value of \( f(x) \) approaches infinity or negative infinity, depending on the direction from which \( x \) approaches.

Therefore, the **range** includes all real numbers except \( f(x) = 0 \), which can be written as:
\[
\text{Range}: \{ f(x) \in \mathbb{R} \mid f(x) \neq 0 \}
\]
or in interval notation:
\[
\text{Range}: (-\infty, 0) \cup (0, +\infty)
\]

### Conclusion:
- **Domain**: \( (-\infty, -2) \cup (-2, +\infty) \)
- **Range**: \( (-\infty, 0) \cup (0, +\infty) \)

Would you like further details or clarifications?

Here are some related questions to expand on this topic:

1. How would the domain and range change for \( f(x) = \frac{1}{x - 3} \)?
2. What happens to the domain and range if the function is \( f(x) = \frac{2}{x + 2} \)?
3. Can a function have an empty domain or range?
4. How does the graph of \( f(x) = \frac{1}{x+2} \) look?
5. What is the domain and range of \( f(x) = \sqrt{x + 2} \)?

**Tip**: Always check for restrictions on the domain, such as division by zero or square roots of negative numbers, to avoid undefined points.

Explore how to find the domain and range of the rational function f(x) = 1 / (x + 2). This problem covers key mathematical concepts such as domain restrictions due to division by zero and properties of rational functions. Suitable for high school students.

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