Question

find the inverse for the function:

\\f(x) = 3x - 21\\

\\f^{-1}(x) = \frac{x}{?} + \\

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Inverse Functions",
"Finding Inverse Algebraically"
],
"new_concepts": [],
"current_concepts": [
"Inverse Functions",
"Finding Inverse Algebraically"
]
}
</pre_analysis>

<reasoning>

Replace f(x) with y

Using the Inverse Functions knowledge point
\[
y = 3x - 21
\]

Swap x and y

Using the Inverse Functions knowledge point
\[
x = 3y - 21
\]

Solve for y

Using the Finding Inverse Algebraically knowledge point
\[

$$\begin{aligned} x + 21 &= 3y \\ y &= \frac{x + 21}{3} \\ y &= \frac{x}{3} + 7 \end{aligned}$$

\]

Express as inverse function

Using the Finding Inverse Algebraically knowledge point
\[
f^{-1}(x) = \frac{x}{3} + 7
\]
</reasoning>

<answer>
Find the inverse for the function:
\(f(x) = 3x - 21\)
\(f^{-1}(x) = \frac{x}{<blank>3</blank>} + <blank>7</blank>\)
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Inverse Functions",
"Finding Inverse Algebraically"
],
"new_concepts": [],
"current_concepts": [
"Inverse Functions",
"Finding Inverse Algebraically"
]
}
</pre_analysis>

<reasoning>

Replace f(x) with y

Using the Inverse Functions knowledge point
\[
y = 3x - 21
\]

Swap x and y

Using the Inverse Functions knowledge point
\[
x = 3y - 21
\]

Solve for y

Using the Finding Inverse Algebraically knowledge point
\[

$$\begin{aligned} x + 21 &= 3y \\ y &= \frac{x + 21}{3} \\ y &= \frac{x}{3} + 7 \end{aligned}$$

\]

Express as inverse function

Using the Finding Inverse Algebraically knowledge point
\[
f^{-1}(x) = \frac{x}{3} + 7
\]
</reasoning>

<answer>
Find the inverse for the function:
\(f(x) = 3x - 21\)
\(f^{-1}(x) = \frac{x}{<blank>3</blank>} + <blank>7</blank>\)
</answer>

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