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The problem asks for the **domain** and **range** of the function \( f(x) = \frac{x+4}{x+2} \).

### 1. Domain:
The domain of a function refers to the set of all possible input values (i.e., values of \(x\)) for which the function is defined.

For a rational function like this, the only restriction is that the denominator cannot be zero, since division by zero is undefined. In this case, the denominator is \(x + 2\). To find when this is zero:

\[
x + 2 = 0
\]
\[
x = -2
\]

Thus, the function is undefined at \(x = -2\), so the domain is all real numbers except \(x = -2\).

**Domain: \( x \in \mathbb{R}, x \neq -2 \)**

### 2. Range:
The range refers to the set of possible output values of the function. To find the range, we first simplify the function and see if there are any restrictions on \(f(x)\).

The function \( f(x) = \frac{x+4}{x+2} \) can be simplified by performing polynomial long division:

\[
\frac{x+4}{x+2} = 1 + \frac{2}{x+2}
\]

Now, observe that for the function to have a certain value, like \( f(x) = 1 \):

\[
1 + \frac{2}{x+2} = 1
\]
\[
\frac{2}{x+2} = 0
\]
\[
2 = 0 \quad (\text{This is impossible!})
\]

So, the function can never be equal to 1. Therefore, the range excludes 1.

**Range: \( f(x) \in \mathbb{R}, f(x) \neq 1 \)**

---

### Summary:
- **Domain:** \( x \in \mathbb{R}, x \neq -2 \)
- **Range:** \( f(x) \in \mathbb{R}, f(x) \neq 1 \)

Would you like further details or explanations?

### Follow-up questions:
1. How can we perform polynomial long division for rational functions?
2. Why is division by zero undefined?
3. How do we identify vertical asymptotes of a rational function?
4. What happens to the graph of \( f(x) \) at \( x = -2 \)?
5. How can we graph the function to visualize the domain and range?

**Tip:** When finding the range of a rational function, simplifying the function and checking for values that make the numerator or denominator zero helps.

The domain and range of the function f(x) = (x + 4)/(x + 2)

Explore how to find the domain and range of the rational function f(x) = (x + 4)/(x + 2). Learn key concepts such as avoiding division by zero and identifying range exclusions using polynomial long division. Suitable for Grades 10-12.

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