Let's find [tex]\((f + g)(x)\)[/tex] by adding the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Given:
[tex]\[ f(x) = 5x + 4 \][/tex]
[tex]\[ g(x) = 7x - 6 \][/tex]
To find [tex]\((f + g)(x)\)[/tex], we add the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f + g)(x) = (5x + 4) + (7x - 6) \][/tex]
Now, let's simplify the expression by combining like terms:
[tex]\[ (f + g)(x) = 5x + 7x + 4 - 6 \][/tex]
[tex]\[ (f + g)(x) = 12x - 2 \][/tex]
So, the simplified form of [tex]\((f + g)(x)\)[/tex] is:
[tex]\[ (f + g)(x) = 12x - 2 \][/tex]
Next, let's consider the domain. Both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are linear functions, meaning they are defined for all real numbers. Since [tex]\((f + g)(x)\)[/tex] is also a linear function, its domain is also all real numbers.
Therefore:
[tex]\[
(f+g)(x) = 12x - 2
\][/tex]
[tex]\[
\text{Domain: all real numbers}
\][/tex]
