To determine the domain and range of the function [tex]\(f(x) = \log(x + 3) - 2\)[/tex], let's go through the problem step-by-step:
### Domain:
1. The function includes a logarithm, [tex]\(\log(x + 3)\)[/tex]. The argument of a logarithmic function must be greater than zero.
2. Set the argument [tex]\(x + 3 > 0\)[/tex].
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x + 3 > 0 \implies x > -3
\][/tex]
4. Therefore, the domain of the function is all values of [tex]\(x\)[/tex] greater than [tex]\(-3\)[/tex]:
[tex]\[
(-3, \infty)
\][/tex]
### Range:
1. The logarithmic function [tex]\(\log(x)\)[/tex] by itself can take any real number. That means the range of [tex]\(\log(x)\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
2. Adding [tex]\(3\)[/tex] inside the logarithm does not affect the range; it translates the function horizontally. Hence, [tex]\(\log(x + 3)\)[/tex] also has a range of [tex]\((-\infty, \infty)\)[/tex].
3. The given function subtracts [tex]\(2\)[/tex] from [tex]\(\log(x + 3)\)[/tex]. Since subtracting a constant will only shift the entire graph downward, it does not impose any limits on the range.
4. Therefore, the range of [tex]\(f(x) = \log(x + 3) - 2\)[/tex] is also all real numbers:
[tex]\[
(-\infty, \infty)
\][/tex]
Thus, the correct answer is:
D. domain: [tex]\((-3, \infty)\)[/tex]; range: [tex]\((-\infty, \infty)\)[/tex]
