To find the domain and range of the function [tex]\( f(x) = \frac{x^2 + 6x + 8}{x + 4} \)[/tex], we'll go through the following steps:
1. Determine the domain:
- The domain of [tex]\( f(x) \)[/tex] is all real numbers [tex]\( x \)[/tex] except where the denominator is zero (since division by zero is undefined).
- Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x + 4 = 0 \implies x = -4
\][/tex]
- Therefore, [tex]\( x \neq -4 \)[/tex]. The domain is all real numbers except [tex]\( x = -4 \)[/tex]:
[tex]\[
\{x \in R \mid x \neq -4\}
\][/tex]
2. Simplify the function:
- Factor the numerator:
[tex]\[
x^2 + 6x + 8 = (x + 2)(x + 4)
\][/tex]
- Now the function becomes:
[tex]\[
f(x) = \frac{(x + 2)(x + 4)}{x + 4}
\][/tex]
- For [tex]\( x \neq -4 \)[/tex], the [tex]\( (x + 4) \)[/tex] terms in the numerator and the denominator cancel each other out:
[tex]\[
f(x) = x + 2, \quad \text{for} \quad x \neq -4
\][/tex]
3. Determine the range:
- Consider the simplified form [tex]\( f(x) = x + 2 \)[/tex] for [tex]\( x \neq -4 \)[/tex].
- The function [tex]\( y = x + 2 \)[/tex] can take any real value for [tex]\( x \)[/tex] in the domain. However, we must exclude the value that occurs when [tex]\( x = -4 \)[/tex]:
[tex]\[
\text{If } x = -4, \quad f(-4) = -4 + 2 = -2
\][/tex]
- Since [tex]\( x = -4 \)[/tex] is not in the domain, [tex]\( y = -2 \)[/tex] is not in the range.
- Therefore, the range is all real numbers except [tex]\( y = -2 \)[/tex]:
[tex]\[
\{y \in R \mid y \neq -2\}
\][/tex]
Putting it all together, the domain and range of the function are:
- Domain: [tex]\(\{x \in R \mid x \neq -4\}\)[/tex]
- Range: [tex]\(\{y \in R \mid y \neq -2\}\)[/tex]
The correct option is:
[tex]\[ D:\{x \in R \mid x \neq -4\}; R:\{y \in R \mid y \neq -2\} \][/tex]
