Answered

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

[tex]\((f+g)(x) = 9x + 3\)[/tex] (Simplify your answer.)

What is the domain of [tex]\(f + g\)[/tex]? Select the correct choice below and, if necessary, fill in the answer box to complete your response.
A. The domain is [tex]\(\{ x \mid \square \}\)[/tex].
(Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B. The domain is [tex]\(\{x \mid x \text{ is any real number} \}\)[/tex].

(b) Find [tex]\((f-g)(x)\)[/tex].

[tex]\((f-g)(x) = \square\)[/tex] (Simplify your answer.)

Sagot :

GinnyAnswer
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]
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