Question

for the given functions f and g, complete parts (a)-(h). for parts (a)-(d), also find the domain.

\\(f(x) = 4x + 1\\); \\(g(x) = 2x - 7\\)

what is the domain of \\(f + g\\)? select the correct choice and, if necessary, fill in the answer box to complete your choice.

a. the domain is \\(\\{x | \square\\}\\).

Explanation:

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<reasoning>

Find the domains of the individual functions

Using the Linear Function Domain knowledge point
\[

$$\begin{aligned} &\text{Domain of } f(x) = 4x + 1 \text{ is } (-\infty, \infty)\\ &\text{Domain of } g(x) = 2x - 7 \text{ is } (-\infty, \infty) \end{aligned}$$

\]

Determine the domain of the sum function

Using the Function Addition and Algebra of Functions knowledge points
\[

$$\begin{aligned} &\text{Domain of } (f+g)(x) = \text{Domain}(f) \cap \text{Domain}(g)\\ &\text{Domain of } f+g = (-\infty, \infty) \cap (-\infty, \infty) = (-\infty, \infty) \end{aligned}$$

\]

Express the domain in set-builder notation

Using the Linear Function Domain knowledge point
\[
\text{Domain} = \{x \mid x \text{ is any real number}\}
\]
</reasoning>

<answer>
What is the domain of \(f + g\)? Select the correct choice and, if necessary, fill in the answer box to complete your choice.

The domain is <blank>\(\{x \mid x \text{ is any real number}\}\)</blank>.
</answer>

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Answer:

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<reasoning>

Find the domains of the individual functions

Using the Linear Function Domain knowledge point
\[

$$\begin{aligned} &\text{Domain of } f(x) = 4x + 1 \text{ is } (-\infty, \infty)\\ &\text{Domain of } g(x) = 2x - 7 \text{ is } (-\infty, \infty) \end{aligned}$$

\]

Determine the domain of the sum function

Using the Function Addition and Algebra of Functions knowledge points
\[

$$\begin{aligned} &\text{Domain of } (f+g)(x) = \text{Domain}(f) \cap \text{Domain}(g)\\ &\text{Domain of } f+g = (-\infty, \infty) \cap (-\infty, \infty) = (-\infty, \infty) \end{aligned}$$

\]

Express the domain in set-builder notation

Using the Linear Function Domain knowledge point
\[
\text{Domain} = \{x \mid x \text{ is any real number}\}
\]
</reasoning>

<answer>
What is the domain of \(f + g\)? Select the correct choice and, if necessary, fill in the answer box to complete your choice.

The domain is <blank>\(\{x \mid x \text{ is any real number}\}\)</blank>.
</answer>

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"Mathematics",
"Algebra",
"Algebra of Functions"
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