Sure, let's solve the problem step by step.
We are given two functions:
[tex]\[ f(x) = -5^x - 4 \][/tex]
[tex]\[ g(x) = -3x - 2 \][/tex]
We need to find the combined function [tex]\((f+g)(x)\)[/tex], which is the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
To do this, we sum the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substitute the given expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f+g)(x) = (-5^x - 4) + (-3x - 2) \][/tex]
Combine the like terms:
[tex]\[ (f+g)(x) = -5^x - 4 - 3x - 2 \][/tex]
Further simplify by combining the constants:
[tex]\[ (f+g)(x) = -5^x - 3x - (4 + 2) \][/tex]
[tex]\[ (f+g)(x) = -5^x - 3x - 6 \][/tex]
Therefore, the function [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ (f+g)(x) = -5^x - 3x - 6 \][/tex]
Comparing this with the provided options, the correct answer is:
B. [tex]\((f+g)(x) = -5^x - 3x - 6\)[/tex]
