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For the given functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex], complete parts (a)-(h). For parts (a)-(d), also find the domain.

[tex]\[ f(x) = 5x + 4 \][/tex]
[tex]\[ g(x) = 7x - 6 \][/tex]

(a) Find [tex]\((f+g)(x)\)[/tex].

[tex]\[ (f+g)(x) = \square \][/tex] (Simplify your answer. Do not factor.)

Sagot :

GinnyAnswer
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]
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