Math Snap

STEP 1

What is this asking?
We need to find all the allowable $x$ values for the function $f(x)$.
Watch out!
Remember, we can't divide by zero!

STEP 2

1. Find the values that make the denominator zero.
2. Exclude those values from the domain.

STEP 3

Alright, so we've got this function, and the only thing we need to watch out for is when the denominator is zero.
So, let's set that denominator, $x^2 + 6x$, equal to zero and solve!
$$x^2 + 6x = 0$$

STEP 4

We can factor out an $x$ from both terms:
$$x(x + 6) = 0$$

STEP 5

Now, we have two factors multiplying to zero.
This means either $x$ is zero or $x + 6$ is zero.
If $x$ is zero, well, $x$ is zero!
If $x + 6$ is zero, we subtract 6 from both sides to get $x = \mathbf{-6}$.

STEP 6

So, we found that $x$ cannot be zero, and $x$ cannot be $-6$.
Any other value of $x$ is perfectly fine!
So, the domain of $f(x)$ is all $x$ except for $\mathbf{0}$ and $\mathbf{-6}$.

The domain of $f$ is $\{x \mid x \neq 0, x \neq -6\}$, which is option B.