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.
