Question

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

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

part 1 of 12

(a) find \\((f + g)(x)\\).

\\((f + g)(x) =\\) (simplify your answer. do not factor.)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Function Addition",
"Algebra of Functions"
],
"current_concepts": [
"Function Addition",
"Algebra of Functions"
]
}
</pre_analysis>

<reasoning>

Identify the given functions

We are given two functions:
\[f(x) = 4x + 1\]
\[g(x) = 2x - 7\]

State the definition of function addition

To find the sum of two functions, we use the definition of Function Addition:
\[(f + g)(x) = f(x) + g(x)\]

Substitute the function expressions

Substitute the given expressions for \(f(x)\) and \(g(x)\) into the addition formula:
\[(f + g)(x) = (4x + 1) + (2x - 7)\]

Combine like terms

Group the variable terms and the constant terms together:
\[(f + g)(x) = (4x + 2x) + (1 - 7)\]
\[(f + g)(x) = 6x - 6\]
</reasoning>

<answer>
(a) Find \((f + g)(x)\).

\((f + g)(x) =\) <blank>\(6x - 6\)</blank> (Simplify your answer. Do not factor.)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Function Addition"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Function Addition",
"Algebra of Functions"
],
"current_concepts": [
"Function Addition",
"Algebra of Functions"
]
}
</pre_analysis>

<reasoning>

Identify the given functions

We are given two functions:
\[f(x) = 4x + 1\]
\[g(x) = 2x - 7\]

State the definition of function addition

To find the sum of two functions, we use the definition of Function Addition:
\[(f + g)(x) = f(x) + g(x)\]

Substitute the function expressions

Substitute the given expressions for \(f(x)\) and \(g(x)\) into the addition formula:
\[(f + g)(x) = (4x + 1) + (2x - 7)\]

Combine like terms

Group the variable terms and the constant terms together:
\[(f + g)(x) = (4x + 2x) + (1 - 7)\]
\[(f + g)(x) = 6x - 6\]
</reasoning>

<answer>
(a) Find \((f + g)(x)\).

\((f + g)(x) =\) <blank>\(6x - 6\)</blank> (Simplify your answer. Do not factor.)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Function Addition"
]
}
</post_analysis>