To solve the problem of finding [tex]\((f - g)(x)\)[/tex] given the functions [tex]\( f(x) = |2x + 3| - 5 \)[/tex] and [tex]\( g(x) = 7 \)[/tex], let's break down the steps:
1. Understand the Definition of the Function [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- [tex]\( f(x) = |2x + 3| - 5 \)[/tex]
- [tex]\( g(x) = 7 \)[/tex]
2. Determine what [tex]\((f - g)(x)\)[/tex] Means:
[tex]\((f - g)(x)\)[/tex] means we subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
3. Substitute the Functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] into the Expression:
We need to substitute the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f - g)(x) = (|2x + 3| - 5) - 7 \][/tex]
4. Simplify the Expression:
Combine the constants:
[tex]\[ (f - g)(x) = |2x + 3| - 5 - 7 \][/tex]
[tex]\[ (f - g)(x) = |2x + 3| - 12 \][/tex]
So, the correct solution is:
[tex]\[ (f - g)(x) = |2x + 3| - 12 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{D. (f - g)(x) = |2 x + 3| - 12} \][/tex]
