Question

1. if \\(f(x) = 3x - 2\\) and \\(g(x) = x^2 + 1\\), find

a.) \\(f(4)\\)
b.) \\(g(-3)\\)
c.) \\(f(2) + g(2)\\)

1. given \\(f(x) = 2x + 5\\) and \\(g(x) = x - 1\\), find

a.) \\((f + g)(x)\\)
b.) \\((f - g)(x)\\)
c.) \\((f \cdot g)(x)\\)

1. if \\(f(x) = x^2 - 4\\) and \\(g(x) = x + 3\\), find \\(\frac{f(x)}{g(x)}\\)

Explanation:

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"Function Operations",
"Function Evaluation",
"Rational Functions"
],
"current_concepts": [
"Function Operations",
"Function Evaluation",
"Rational Functions"
]
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</pre_analysis>

<reasoning>
The image is oriented upside down. Let's first transcribe the three questions accurately by rotating the text mentally:

Question 1:
If \(f(x) = 3x - 2\) and \(g(x) = x^2 + 1\), find:
a.) \(f(4)\)
b.) \(g(-3)\)
c.) \(f(2) + g(2)\)

Question 2:
Given \(f(x) = 2x + 5\) and \(g(x) = x - 1\), find:
a.) \((f + g)(x)\)
b.) \((f - g)(x)\)
c.) \((f \cdot g)(x)\)

Question 3:
If \(f(x) = x^2 - 4\) and \(g(x) = x + 3\), find \(\frac{f(x)}{g(x)}\).

Let's solve each question step-by-step.

Evaluate functions for Question 1

We are given \(f(x) = 3x - 2\) and \(g(x) = x^2 + 1\).

For part a.), substitute \(x = 4\) into \(f(x)\):
\[
f(4) = 3(4) - 2 = 12 - 2 = 10
\]

For part b.), substitute \(x = -3\) into \(g(x)\):
\[
g(-3) = (-3)^2 + 1 = 9 + 1 = 10
\]

For part c.), first find \(f(2)\) and \(g(2)\):
\[
f(2) = 3(2) - 2 = 6 - 2 = 4
\]
\[
g(2) = (2)^2 + 1 = 4 + 1 = 5
\]
Then add them together:
\[
f(2) + g(2) = 4 + 5 = 9
\]

Perform operations for Question 2

We are given \(f(x) = 2x + 5\) and \(g(x) = x - 1\).

For part a.), find the sum function:
\[
(f + g)(x) = f(x) + g(x) = (2x + 5) + (x - 1) = 3x + 4
\]

For part b.), find the difference function:
\[
(f - g)(x) = f(x) - g(x) = (2x + 5) - (x - 1) = 2x + 5 - x + 1 = x + 6
\]

For part c.), find the product function:
\[
(f \cdot g)(x) = f(x) \cdot g(x) = (2x + 5)(x - 1) = 2x^2 - 2x + 5x - 5 = 2x^2 + 3x - 5
\]

Find the quotient for Question 3

We are given \(f(x) = x^2 - 4\) and \(g(x) = x + 3\).

Find the quotient function \(\frac{f(x)}{g(x)}\):
\[
\frac{f(x)}{g(x)} = \frac{x^2 - 4}{x + 3}
\]
Since \(x^2 - 4 = (x - 2)(x + 2)\), there are no common factors with the denominator \(x + 3\) to simplify further. Thus, the expression is:
\[
\frac{x^2 - 4}{x + 3} \quad (\text{where } x
eq -3)
\]
</reasoning>

<answer>

Question 1

If \(f(x) = 3x - 2\) and \(g(x) = x^2 + 1\):

• a.) \(f(4) = 10\)
• b.) \(g(-3) = 10\)
• c.) \(f(2) + g(2) = 9\)

Question 2

Given \(f(x) = 2x + 5\) and \(g(x) = x - 1\):

• a.) \((f + g)(x) = 3x + 4\)
• b.) \((f - g)(x) = x + 6\)
• c.) \((f \cdot g)(x) = 2x^2 + 3x - 5\)

Question 3

If \(f(x) = x^2 - 4\) and \(g(x) = x + 3\):
\[
\frac{f(x)}{g(x)} = \frac{x^2 - 4}{x + 3} \quad (x
eq -3)
\]
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Function Operations"
]
}
</post_analysis>

Answer:

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"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Function Operations",
"Function Evaluation",
"Rational Functions"
],
"current_concepts": [
"Function Operations",
"Function Evaluation",
"Rational Functions"
]
}
</pre_analysis>

<reasoning>
The image is oriented upside down. Let's first transcribe the three questions accurately by rotating the text mentally:

Question 1:
If \(f(x) = 3x - 2\) and \(g(x) = x^2 + 1\), find:
a.) \(f(4)\)
b.) \(g(-3)\)
c.) \(f(2) + g(2)\)

Question 2:
Given \(f(x) = 2x + 5\) and \(g(x) = x - 1\), find:
a.) \((f + g)(x)\)
b.) \((f - g)(x)\)
c.) \((f \cdot g)(x)\)

Question 3:
If \(f(x) = x^2 - 4\) and \(g(x) = x + 3\), find \(\frac{f(x)}{g(x)}\).

Let's solve each question step-by-step.

Evaluate functions for Question 1

We are given \(f(x) = 3x - 2\) and \(g(x) = x^2 + 1\).

For part a.), substitute \(x = 4\) into \(f(x)\):
\[
f(4) = 3(4) - 2 = 12 - 2 = 10
\]

For part b.), substitute \(x = -3\) into \(g(x)\):
\[
g(-3) = (-3)^2 + 1 = 9 + 1 = 10
\]

For part c.), first find \(f(2)\) and \(g(2)\):
\[
f(2) = 3(2) - 2 = 6 - 2 = 4
\]
\[
g(2) = (2)^2 + 1 = 4 + 1 = 5
\]
Then add them together:
\[
f(2) + g(2) = 4 + 5 = 9
\]

Perform operations for Question 2

We are given \(f(x) = 2x + 5\) and \(g(x) = x - 1\).

For part a.), find the sum function:
\[
(f + g)(x) = f(x) + g(x) = (2x + 5) + (x - 1) = 3x + 4
\]

For part b.), find the difference function:
\[
(f - g)(x) = f(x) - g(x) = (2x + 5) - (x - 1) = 2x + 5 - x + 1 = x + 6
\]

For part c.), find the product function:
\[
(f \cdot g)(x) = f(x) \cdot g(x) = (2x + 5)(x - 1) = 2x^2 - 2x + 5x - 5 = 2x^2 + 3x - 5
\]

Find the quotient for Question 3

We are given \(f(x) = x^2 - 4\) and \(g(x) = x + 3\).

Find the quotient function \(\frac{f(x)}{g(x)}\):
\[
\frac{f(x)}{g(x)} = \frac{x^2 - 4}{x + 3}
\]
Since \(x^2 - 4 = (x - 2)(x + 2)\), there are no common factors with the denominator \(x + 3\) to simplify further. Thus, the expression is:
\[
\frac{x^2 - 4}{x + 3} \quad (\text{where } x
eq -3)
\]
</reasoning>

<answer>

Question 1

If \(f(x) = 3x - 2\) and \(g(x) = x^2 + 1\):

• a.) \(f(4) = 10\)
• b.) \(g(-3) = 10\)
• c.) \(f(2) + g(2) = 9\)

Question 2

Given \(f(x) = 2x + 5\) and \(g(x) = x - 1\):

• a.) \((f + g)(x) = 3x + 4\)
• b.) \((f - g)(x) = x + 6\)
• c.) \((f \cdot g)(x) = 2x^2 + 3x - 5\)

Question 3

If \(f(x) = x^2 - 4\) and \(g(x) = x + 3\):
\[
\frac{f(x)}{g(x)} = \frac{x^2 - 4}{x + 3} \quad (x
eq -3)
\]
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Function Operations"
]
}
</post_analysis>