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.