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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]\( f(x) = x - 9 \)[/tex]
[tex]\( g(x) = 6x^2 \)[/tex]

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

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



Answer :

GinnyAnswer
To solve this, we need to determine [tex]\((f+g)(x)\)[/tex], which represents the sum of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].

Given:

[tex]\[ f(x) = x - 9 \][/tex]
[tex]\[ g(x) = 6x^2 \][/tex]

The combined function [tex]\((f+g)(x)\)[/tex] is found by summing [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:

[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]

Substituting the given expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:

[tex]\[ (f+g)(x) = (x - 9) + (6x^2) \][/tex]

Combining like terms, we get:

[tex]\[ (f+g)(x) = 6x^2 + x - 9 \][/tex]

So, the simplified form of [tex]\((f+g)(x)\)[/tex] is:

[tex]\[ (f+g)(x) = 6x^2 + x - 9 \][/tex]

Now, we need to determine the domain of the function [tex]\((f+g)(x)\)[/tex]. The individual functions [tex]\(f(x) = x - 9\)[/tex] and [tex]\(g(x) = 6x^2\)[/tex] are both defined for all real numbers, since there are no restrictions such as division by zero or square roots of negative numbers.

Therefore, the domain of [tex]\((f+g)(x)\)[/tex] is:

[tex]\[ \text{Domain} = \text{all real numbers} \][/tex]

In summary:

[tex]\[ (f+g)(x) = 6x^2 + x - 9 \][/tex]

The domain is all real numbers.

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