To find the domain and range of the function \( f(x) = \frac{x^2 - 3x - 28}{x + 4} \), we need to analyze the function step by step.
Step 1: Find the Domain
The function \( f(x) \) is a rational function, which means it is defined for all real numbers except where the denominator is zero.
Setting the denominator equal to zero: \[ x + 4 = 0 \implies x = -4 \]
Thus, the domain of \( f(x) \) is all real numbers except \( -4 \): \[ \text{Domain: } x \in \mathbb{R}, x \neq -4 \]
Step 2: Find the Range
To find the range, we can simplify the function if possible. First, we can factor the numerator \( x^2 - 3x - 28 \).
We can look for two numbers that multiply to \(-28\) and add to \(-3\). The numbers are \( -7 \) and \( 4 \). Thus, we can factor it as: \[ x^2 - 3x - 28 = (x - 7)(x + 4) \]
Now substituting this back into the function gives: \[ f(x) = \frac{(x - 7)(x + 4)}{x + 4} \]
For \( x \neq -4 \), we can simplify: \[ f(x) = x - 7 \quad (x \neq -4) \]
Identifying the behavior near the excluded value
As \( x \) approaches \( -4 \), the simplified function \( f(x) \) approaches: \[ f(-4) = -4 - 7 = -11 \] But \( f(-4) \) is undefined since \( x = -4 \) is in the denominator of the original function.
Conclusion
Since \( f(x) = x - 7 \) is a linear function (and hence covers all real numbers) except at the point \( y = -11 \) (the value that \( f(x) \) approaches as \( x \) approaches \(-4\)), the range is all real numbers except \(-11\):
Thus, we conclude:
- Domain: \( x \in \mathbb{R}, x \neq -4 \)
- Range: \( y \in \mathbb{R}, y \neq -11 \)
So, the correct choice among the given options is: Domain is \( x \) is all real numbers where \( x \) does not equal \(-4\), range is \( y \) is all real numbers where \( y \) does not equal \(-11\).