Math Snap

STEP 1

1. The function $f(x) = \frac{3}{x+4}$ is a rational function.
2. The domain of a function is the set of all possible input values (x-values) that do not make the function undefined.
3. The range of a function is the set of all possible output values (y-values) that the function can take.

STEP 2

1. Determine the domain of $f(x)$.
2. Determine the range of $f(x)$.

STEP 3

To find the domain of $f(x) = \frac{3}{x+4}$, identify the values of $x$ that make the denominator zero, as these will make the function undefined.

STEP 4

Set the denominator equal to zero: $x + 4 = 0$.

STEP 5

Solve for $x$: $x = -4$.

STEP 6

The domain of $f(x)$ is all real numbers except $x = -4$. In interval notation, this is $(-\infty, -4) \cup (-4, \infty)$.

STEP 7

To determine the range of $f(x) = \frac{3}{x+4}$, consider the behavior of the function as $x$ approaches the vertical asymptote $x = -4$ and the horizontal asymptote $y = 0$.

STEP 8

As $x \to -4^+$ or $x \to -4^-$, $f(x)$ approaches $\pm \infty$, indicating that $y$ can take any real value except the horizontal asymptote value.

STEP 9

The horizontal asymptote at $y = 0$ suggests that $f(x)$ never actually reaches $y = 0$.

Thus, the range of $f(x)$ is all real numbers except $y = 0$. In interval notation, this is $(-\infty, 0) \cup (0, \infty)$.
The domain of $f(x)$ is $(-\infty, -4) \cup (-4, \infty)$.
The range of $f(x)$ is $(-\infty, 0) \cup (0, \infty)$.