Let's consider the function $f$ defined by the formula:

$f(x) = \frac{3x + 6}{x - 2}$

The problem is asking to find the domain of $f$.

Finding the Domain of $f$

The domain of a function consists of all the possible values of $x$ that can be input into the function without causing any undefined operations. For rational functions like this one, we need to ensure that the denominator is not zero because division by zero is undefined.

The denominator of $f(x)$ is $x - 2$. To find the values of $x$ that are not allowed, we set the denominator equal to zero and solve for $x$:

$x - 2 = 0$

$x = 2$

Thus, $x = 2$ is the value that will make the denominator zero and therefore is not in the domain of $f$.

Domain of $f$

The domain of $f$ includes all real numbers except $x = 2$. In interval notation, the domain is:

$(-\infty, 2) \cup (2, \infty)$

So, the domain of $f(x) = \frac{3x + 6}{x - 2}$ is $\mathbb{R} \setminus \{ 2 \}$.

Would you like further details or have any questions?

Related Questions

1. How do you determine the vertical asymptotes of a rational function?
2. What are the horizontal asymptotes of the function $f(x) = \frac{3x + 6}{x - 2}$?
3. How do you find the x-intercepts and y-intercepts of the function $f(x)$?
4. Can you simplify the expression $f(x) = \frac{3x + 6}{x - 2}$?
5. How would you sketch the graph of $f(x) = \frac{3x + 6}{x - 2}$?

Tip

When analyzing the domain of a rational function, always check for values that make the denominator zero, as these are the points where the function is undefined.