To determine which function defines [tex]\((f+g)(x)\)[/tex], we first need to understand the individual functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
Given:
[tex]\[ f(x) = \ln(2x) - 15 \][/tex]
[tex]\[ g(x) = 3 \sin(x) - 7 \][/tex]
To find [tex]\((f+g)(x)\)[/tex], we simply add [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f+g)(x) = f(x) + g(x)
\][/tex]
Substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f+g)(x) = (\ln(2x) - 15) + (3 \sin(x) - 7)
\][/tex]
Combine the terms:
[tex]\[
(f+g)(x) = \ln(2x) - 15 + 3 \sin(x) - 7
\][/tex]
Simplify the expression by combining the constants:
[tex]\[
(f+g)(x) = \ln(2x) + 3 \sin(x) - 15 - 7
\][/tex]
[tex]\[
(f+g)(x) = \ln(2x) + 3 \sin(x) - 22
\][/tex]
So, the function that defines [tex]\((f+g)(x)\)[/tex] is:
[tex]\[
(f+g)(x) = \ln(2x) + 3 \sin(x) - 22
\][/tex]
The correct answer is:
B. [tex]\((f+g)(x) = \ln(2x) + 3 \sin(x) - 22\)[/tex]
