Let's solve each part of the question step-by-step.
### (a) Find [tex]\((f + g)(x)\)[/tex]
Given:
[tex]\[ (f + g)(x) = 9x + 3 \][/tex]
The expression [tex]\((f + g)(x)\)[/tex] is already simplified since it is given as [tex]\(9x + 3\)[/tex].
To check the domain of [tex]\((f + g)\)[/tex]:
Since [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are linear functions and their sum [tex]\(9x + 3\)[/tex] is also a linear function, the domain of a linear function is all real numbers.
Thus, the domain of [tex]\((f + g)\)[/tex] is:
B. The domain is [tex]\(\{ x \mid x \text{ is any real number} \}\)[/tex]
### (b) Find [tex]\((f - g)(x)\)[/tex]
To find [tex]\((f - g)(x)\)[/tex]:
Given that [tex]\(f(x) = 4.5x + 1.5\)[/tex] and [tex]\(g(x) = 4.5x + 1.5\)[/tex] (determined from the sum provided in the question),
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
Substituting the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f - g)(x) = (4.5x + 1.5) - (4.5x + 1.5)
\][/tex]
Simplifying this expression:
[tex]\[
(f - g)(x) = 4.5x + 1.5 - 4.5x - 1.5
\][/tex]
[tex]\[
(f - g)(x) = 0
\][/tex]
Thus, the simplified form of [tex]\((f - g)(x)\)[/tex] is:
[tex]\((f - g)(x) = 0\)[/tex]
### Summary
1. [tex]\((f + g)(x) = 9x + 3\)[/tex]
2. The domain of [tex]\((f + g)\)[/tex] is [tex]\(\{ x \mid x \text{ is any real number} \}\)[/tex]
3. [tex]\((f - g)(x) = 0\)[/tex]
