instruction_seed stringlengths 17 6.02k | response_seed stringlengths 17 6.02k | reasoning stringlengths 1.28k 104k | deepseek_solution stringlengths 7 3.7k | __original_row_idx int64 0 40k | source stringclasses 1
value | final_reasoning_trace stringlengths 1.61k 106k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
Find the range of the function
\[g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}}.\] | Find the range of the function
\[g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}}.\] | Alright, let's try to figure out the range of this function \( g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}} \). Hmm, okay, so the range is all the possible y-values that this function can take. To find the range, I need to determine all real numbers y such that there exists some x where \( y = \frac{2x + 1}{\sqrt{x^2 + 4x... | To find the range of the function \( g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}} \), we start by determining the domain. The expression inside the square root, \( x^2 + 4x + 1 \), must be positive. Solving the quadratic inequality, we find the domain is \( (-\infty, -2 - \sqrt{3}) \cup (-2 + \sqrt{3}, \infty) \).
Next, ... | 10,835 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's try to figure out the range of this function \( g(x) = \frac{2x + 1}{\sqrt{x^2 + 4x + 1}} \). Hmm, okay, so the range is all the possible y-values that this function can take. To find the range, I need to determine all real numbers y such that there exists some x where \( y = \frac{2x + 1}{\sqrt{... | [
{
"from": "human",
"value": "Find the range of the function\n\\[g(x) = \\frac{2x + 1}{\\sqrt{x^2 + 4x + 1}}.\\]"
},
{
"from": "gpt",
"value": "<think>\nAlright, let's try to figure out the range of this function \\( g(x) = \\frac{2x + 1}{\\sqrt{x^2 + 4x + 1}} \\). Hmm, okay, so the range is all ... |
Find the range of the function $f(x) = \arctan\left(\frac{x^3-x}{x^2+1}\right)$, where $x$ is any real number. | Find the range of the function $f(x) = \arctan\left(\frac{x^3-x}{x^2+1}\right)$, where $x$ is any real number. | Okay, so I need to find the range of the function \( f(x) = \arctan\left(\frac{x^3 - x}{x^2 + 1}\right) \). Hmm, the range of an arctan function is usually between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), right? But since the argument here is another function \( \frac{x^3 - x}{x^2 + 1} \), I need to figure out wha... | To find the range of the function \( f(x) = \arctan\left(\frac{x^3 - x}{x^2 + 1}\right) \), we need to analyze the inner function \( g(x) = \frac{x^3 - x}{x^2 + 1} \).
First, we simplify \( g(x) \):
\[
g(x) = \frac{x^3 - x}{x^2 + 1} = x - \frac{2x}{x^2 + 1}
\]
Next, we analyze the behavior of \( g(x) \):
- As \( x \t... | 2,042 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the range of the function \( f(x) = \arctan\left(\frac{x^3 - x}{x^2 + 1}\right) \). Hmm, the range of an arctan function is usually between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), right? But since the argument here is another function \( \frac{x^3 - x}{x^2 + 1} \), I need to figure... | [
{
"from": "human",
"value": "Find the range of the function $f(x) = \\arctan\\left(\\frac{x^3-x}{x^2+1}\\right)$, where $x$ is any real number."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the range of the function \\( f(x) = \\arctan\\left(\\frac{x^3 - x}{x^2 + 1}\\right) \\)... |
"Find the volume of the largest cone that can be inscribed in a right square pyramid with base edges(...TRUNCATED) | "Find the volume of the largest cone that can be inscribed in a right square pyramid with base edges(...TRUNCATED) | "Okay, so I need to find the volume of the largest cone that can be inscribed in a right square pyra(...TRUNCATED) | "To find the volume of the largest cone that can be inscribed in a right square pyramid with base ed(...TRUNCATED) | 14,593 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find the volume of the largest cone that can be inscribed in a right sq(...TRUNCATED) | [{"from":"human","value":"Find the volume of the largest cone that can be inscribed in a right squar(...TRUNCATED) |
"Problem:\nA water tank can hold 1000 liters of water. Water flows into the tank at a rate of 2 lite(...TRUNCATED) | "Problem:\nA water tank can hold 1000 liters of water. Water flows into the tank at a rate of 2 lite(...TRUNCATED) | "Okay, so there's this problem about a water tank, right? Let me try to figure it out step by step. (...TRUNCATED) | "To determine how long it takes to fill the 1000-liter tank, we first calculate the net inflow rate (...TRUNCATED) | 19,453 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so there's this problem about a water tank, right? Let me try to figure it out step (...TRUNCATED) | [{"from":"human","value":"Problem:\nA water tank can hold 1000 liters of water. Water flows into the(...TRUNCATED) |
"A right circular cone has a slant height of $15$ and a base radius of $6$. Determine the total surf(...TRUNCATED) | "A right circular cone has a slant height of $15$ and a base radius of $6$. Determine the total surf(...TRUNCATED) | "Okay, so I need to find the total surface area of a right circular cone. The problem gives me the s(...TRUNCATED) | "To determine the total surface area of a right circular cone with a slant height of 15 and a base r(...TRUNCATED) | 19,439 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find the total surface area of a right circular cone. The problem gives(...TRUNCATED) | [{"from":"human","value":"A right circular cone has a slant height of $15$ and a base radius of $6$.(...TRUNCATED) |
"Let a sequence of consecutive natural numbers be given. Determine $n$ if the sum of all the terms i(...TRUNCATED) | "Let a sequence of consecutive natural numbers be given. Determine $n$ if the sum of all the terms i(...TRUNCATED) | "Okay, so I have this problem here: Given a sequence of consecutive natural numbers, determine n if (...TRUNCATED) | "Given a sequence of consecutive natural numbers, we need to determine \\( n \\) such that the sum o(...TRUNCATED) | 16,236 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I have this problem here: Given a sequence of consecutive natural numbers, determ(...TRUNCATED) | [{"from":"human","value":"Let a sequence of consecutive natural numbers be given. Determine $n$ if t(...TRUNCATED) |
"Let $p(x)$ be a monic cubic polynomial such that $p(1) = 3$, $p(2) = 4$, $p(3) = 5$, and $p(4) = 6$(...TRUNCATED) | "Let $p(x)$ be a monic cubic polynomial such that $p(1) = 3$, $p(2) = 4$, $p(3) = 5$, and $p(4) = 6$(...TRUNCATED) | "Okay, so I need to find the remainder when the sum of p(1) to p(100) is divided by 101, where p(x) (...TRUNCATED) | "Given a monic cubic polynomial \\( p(x) \\) such that \\( p(1) = 3 \\), \\( p(2) = 4 \\), \\( p(3) (...TRUNCATED) | 31,416 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find the remainder when the sum of p(1) to p(100) is divided by 101, wh(...TRUNCATED) | [{"from":"human","value":"Let $p(x)$ be a monic cubic polynomial such that $p(1) = 3$, $p(2) = 4$, $(...TRUNCATED) |
"A regular polyhedron has faces that are regular polygons, and each face is surrounded by exactly 5 (...TRUNCATED) | "A regular polyhedron has faces that are regular polygons, and each face is surrounded by exactly 5 (...TRUNCATED) | "Okay, so I need to figure out the minimum number of edges a regular polyhedron can have if each fac(...TRUNCATED) | "To determine the minimum number of edges a regular polyhedron can have if each face is surrounded b(...TRUNCATED) | 3,985 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to figure out the minimum number of edges a regular polyhedron can have if(...TRUNCATED) | [{"from":"human","value":"A regular polyhedron has faces that are regular polygons, and each face is(...TRUNCATED) |
What is the product of the natural-number factors of 8 that are also perfect squares? | What is the product of the natural-number factors of 8 that are also perfect squares? | "Okay, let's see. The question is asking for the product of the natural-number factors of 8 that are(...TRUNCATED) | "To find the product of the natural-number factors of 8 that are also perfect squares, we start by i(...TRUNCATED) | 15,645 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, let's see. The question is asking for the product of the natural-number factors of 8(...TRUNCATED) | [{"from":"human","value":"What is the product of the natural-number factors of 8 that are also perfe(...TRUNCATED) |
"On the Cartesian plane, the equation of a conic section is given by\n\\[\\frac{x^2}{3^2} + \\frac{y(...TRUNCATED) | "On the Cartesian plane, the equation of a conic section is given by\n\\[\\frac{x^2}{3^2} + \\frac{y(...TRUNCATED) | "Okay, let me try to figure out this problem step by step. So, we have a conic section given by the (...TRUNCATED) | "The given equation of the conic section is \\(\\frac{x^2}{3^2} + \\frac{y^2}{4^2} = 1\\), which rep(...TRUNCATED) | 12,737 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, let me try to figure out this problem step by step. So, we have a conic section give(...TRUNCATED) | [{"from":"human","value":"On the Cartesian plane, the equation of a conic section is given by\n\\[\\(...TRUNCATED) |
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