messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-5(-1cos(1π(2|x + 1| - 3) - 2))^2 + 1(-1cos(1π(2|x + 1| - 3) - 2)) + 1) - 2)/(2(-5(-1cos(1π(2|x + 1| - 3) - 2))^2 + 1(-1cos(1π(2|x + 1| - 3) - 2)) + 1) - 3) and g(x) = 1(-5(-3sin(1πx - 2)) + 3)^2 + 1(-5(-3sin(1πx - 2)) + 3) - 4 intersect in the int... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(2π(3|((2x + 2)/(-1x - 2)) + 1| - 3) + 3) + 3 and g(x) = 2((-2x - 2)/(-1x + 2)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "How many solutions does the equation (1(-1cos(2π(-2x - 5)) - 3) - 1)/(3(-1cos(2π(-2x - 5)) - 3) + 1) = -2((-1x - 3)/(-2x + 1))^2 + 5((-1x - 3)/(-2x + 1)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((1(-3|(1sin(2πx) + 2) - 2| + 2) + 3)/(-3(-3|(1sin(2πx) + 2) - 2| + 2) + 2))^2 + 4((1(-3|(1sin(2πx) + 2) - 2| + 2) + 3)/(-3(-3|(1sin(2πx) + 2) - 2| + 2) + 2)) and g(x) = -3((1(1x + 3))/(-1(1x + 3) - 2))^2 - 1((1(1x + 3))/(-1(1x +... | 35 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-1|(-4(-2e^(1x) + 3) - 1) - 1| - 3) - 3)/(2(-1|(-4(-2e^(1x) + 3) - 1) - 1| - 3)) and g(x) = (-1(-2(3cos(1πx - 1) - 2)^2 - 3(3cos(1πx - 1) - 2) + 3) + 3)/(3(-2(3cos(1πx - 1) - 2)^2 - 3(3cos(1πx - 1) - 2) + 3) - 2) intersect in the interval [-10, 10].... | 40 | MATH |
[
{
"content": "How many solutions does the equation -1cos(2π(-2|(-1x^2 - 4) + 3|) + 2) - 3 = -3(1|x + 2|)^2 + 3(1|x + 2|) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(2(2sin(2π(1x^2 + 2x + 2) - 3) + 1) + 1)| + 1 and g(x) = 1|((-1(-4x - 1))/(1(-4x - 1) - 3)) + 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 82 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-2(3|(-3sin(2πx) - 3) - 3| + 2)^2 + 1(3|(-3sin(2πx) - 3) - 3| + 2) - 5) - 3)/(-2(-2(3|(-3sin(2πx) - 3) - 3| + 2)^2 + 1(3|(-3sin(2πx) - 3) - 3| + 2) - 5) - 2) and g(x) = (-1(-1sin(2π(-4x^2 + 3x + 4) - 2) + 1))/(2(-1sin(2π(-4x^2 ... | 811 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-1|(5((-2x - 1)/(2x - 1)) - 1) + 2| + 1)^2 - 3(-1|(5((-2x - 1)/(2x - 1)) - 1) + 2| + 1) + 1 and g(x) = (-1(1cos(1π(3|x - 3| + 1) + 1)) + 1)/(-2(1cos(1π(3|x - 3| + 1) + 1)) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in... | 57 | MATH |
[
{
"content": "How many solutions does the equation 2|(-5((-3(1x) - 2)/(-2(1x) - 3))^2 - 4((-3(1x) - 2)/(-2(1x) - 3)) + 5)| - 1 = -1(2sin(2π((1x)/(-3x + 2))) - 1) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1e^(-1(-3(-3|((2x + 1)/(-2x - 3)) + 3| - 2) + 4)) and g(x) = -3|(2sin(3π(1x - 4) + 2) + 1)| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(2π((2(1(1x - 4)^2 - 5(1x - 4) - 5) + 3)/(1(1(1x - 4)^2 - 5(1x - 4) - 5) + 3)) + 2) - 2 and g(x) = 2e^(-1(3|(-2x + 1)| - 3)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 41 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(-2(2|((-3x - 2)/(-1x - 3)) - 3| - 3) + 3) + 3) and g(x) = -1|(4(2cos(2πx) + 1)^2 + 5(2cos(2πx) + 1) - 1) - 2| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 25 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(5(3(-1|x - 2| - 1) - 5)^2 + 3(3(-1|x - 2| - 1) - 5) - 4))/(2(5(3(-1|x - 2| - 1) - 5)^2 + 3(3(-1|x - 2| - 1) - 5) - 4) - 1) and g(x) = 1|((-1(-1sin(1πx - 3) + 2) + 3)/(-1(-1sin(1πx - 3) + 2) + 2))| - 1 where -10 ≤ x ≤ 10.\n... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(2π(2|(-5(4x^2 - 2x + 3) - 2) + 1|) - 2) + 1 and g(x) = 2|(-1sin(1π(-4x^2 - 1x - 5))) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 366 | MATH |
[
{
"content": "How many solutions does the equation 3|((2(1sin(2π(4x^2 - 3x + 1) + 3) + 1) - 2)/(-3(1sin(2π(4x^2 - 3x + 1) + 3) + 1) - 2)) - 2| + 3 = 5(-2sin(2π((3x + 1)/(1x + 2)) - 1) - 2)^2 - 3(-2sin(2π((3x + 1)/(1x + 2)) - 1) - 2) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format... | 164 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-2cos(2π(-3x - 4) + 2) + 3) + 2| + 2 and g(x) = -3(-1sin(3πx - 2) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(1π(-3(-5x^2 - 2x + 1) + 2) + 3) and g(x) = -2|(1x - 5) - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 118 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(3cos(2π(-1(1|x + 2| + 2)^2 + 1(1|x + 2| + 2) + 4)) - 2) and g(x) = (-3(-4(-3x^2 + 4x + 3) - 5) - 1)/(1(-4(-3x^2 + 4x + 3) - 5) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 536 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(2|(-5x^2 + 4x + 4)| + 1) + 5 and g(x) = -4(-3cos(3πx - 2) - 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation 3|(-2(2(1cos(1πx + 2) + 1))^2 + 3(2(1cos(1πx + 2) + 1)) + 2) + 3| + 1 = 2e^(-2((-1(-2x^2 + 5x + 3) - 2)/(1(-2x^2 + 5x + 3) + 2))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-3e^(1(-5x^2 + 1x + 2)) - 1) + 1| + 1 and g(x) = -3cos(3π(3|x + 1| + 3) - 3) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 164 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-3cos(2π(-2x^2 + 1x + 4)) - 2) + 1)/(-3(-3cos(2π(-2x^2 + 1x + 4)) - 2) + 1) and g(x) = 1|((1x - 1)/(-3x - 2)) + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 112 | MATH |
[
{
"content": "How many solutions does the equation 3e^(1(-1|((-3(-4x - 1) + 1)/(3(-4x - 1) - 2)) + 1|)) + 1 = 1|(2cos(3π((-2x - 3)/(2x - 1)) - 1) + 2) - 1| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 88 | MATH |
[
{
"content": "How many solutions does the equation -3e^(1(3(-3|x + 1|) + 1)) = -2sin(1π(1x^2 + 5x + 3) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 214 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-2cos(1π((1x)/(-3x - 3)) + 1) - 3) + 4 and g(x) = (1(2cos(2πx + 2) - 1) + 2)/(3(2cos(2πx + 2) - 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(1(-1|(-1x - 1)| - 2)^2 + 3(-1|(-1x - 1)| - 2) + 1) + 2)/(3(1(-1|(-1x - 1)| - 2)^2 + 3(-1|(-1x - 1)| - 2) + 1) - 2) and g(x) = -2sin(3π(-5(-3|x - 1|) - 5)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\bo... | 720 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-2((-2(2|x| - 2) + 2)/(-2(2|x| - 2) - 1))^2 + 2((-2(2|x| - 2) + 2)/(-2(2|x| - 2) - 1)) + 2) - 3 and g(x) = -2sin(2π((1(-4x + 4) - 3)/(2(-4x + 4) + 2)) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed... | 9 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(1π(2|(2x + 5)| - 1) + 3) + 2 and g(x) = -3cos(2π(-1|x + 1| - 3) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(2sin(2π(-1((2x + 1)/(1x - 2)) - 1) - 3) + 3) - 2| + 3 and g(x) = -3cos(1π(3(-2|x + 3| + 1)^2 - 2(-2|x + 3| + 1) - 5) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 721 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(3cos(3π(-3|(-4x + 2) + 3| - 2)) + 2)^2 + 2(3cos(3π(-3|(-4x + 2) + 3| - 2)) + 2) + 1 and g(x) = -1((-1(-2cos(1πx + 1) - 2) - 2)/(-2(-2cos(1πx + 1) - 2) - 2)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your... | 1146 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(5(1sin(1πx) + 2)^2 - 2(1sin(1πx) + 2))| - 1 and g(x) = (-1(-2sin(3πx - 1) - 1) - 1)/(1(-2sin(3πx - 1) - 1)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "How many solutions does the equation -1|((2(-2cos(2π(5x + 2) + 3) + 1) - 2)/(2(-2cos(2π(5x + 2) + 3) + 1) - 1)) - 3| - 2 = -3sin(3π(-2((3x - 1)/(-3x - 3)) + 3) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 85 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|((-3(2sin(1πx - 2) + 2) - 3)/(-2(2sin(1πx - 2) + 2) + 3)) - 1| + 1 and g(x) = -3cos(2π(-3x^2 - 1x + 4) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 310 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(1(-4((1x + 1)/(1x + 3))^2 + 2((1x + 1)/(1x + 3)) - 2) + 2) + 1| - 3 and g(x) = -3(1cos(3π(3|x|) + 2) - 2)^2 + 4(1cos(3π(3|x|) + 2) - 2) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 22 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(1π(-4(-2|(-3x^2 - 3x + 4) - 3| + 2) - 1) - 2) - 2 and g(x) = -1e^(-1(2|((1x + 3)/(1x))| + 1)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1370 | MATH |
[
{
"content": "How many solutions does the equation 3|((1(-1x^2 - 4x) + 3)/(-2(-1x^2 - 4x) + 3)) - 2| - 1 = 2|(-3x^2 + 4x + 3) - 1| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(4((2(3|x - 3|) - 3)/(-2(3|x - 3|) + 3)))^2 - 5(4((2(3|x - 3|) - 3)/(-2(3|x - 3|) + 3))) - 4 and g(x) = (-3(1sin(3π(-1|x + 1| + 3))) - 1)/(3(1sin(3π(-1|x + 1| + 3))) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\b... | 62 | MATH |
[
{
"content": "How many solutions does the equation (1(-2|(1e^(1x) - 1) - 3| + 3) - 1)/(-2(-2|(1e^(1x) - 1) - 3| + 3) + 3) = 3cos(2π(-2|x - 2| + 2) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many solutions does the equation (1(-1(4x^2 + 1x) + 4))/(-1(-1(4x^2 + 1x) + 4) + 2) = -1cos(3π(-3x^2 + 5x + 5) + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(3(1|((2x + 1)/(-1x - 3)) - 1| - 2)^2 - 2(1|((2x + 1)/(-1x - 3)) - 1| - 2) + 4) + 3 and g(x) = 5(3cos(1π((-1x - 1)/(-2x + 1)) - 2) + 1)^2 - 2(3cos(1π((-1x - 1)/(-2x + 1)) - 2) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer i... | 45 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(1|((2(1x - 2) - 1)/(-2(1x - 2) + 3)) - 3|)^2 + 3(1|((2(1x - 2) - 1)/(-2(1x - 2) + 3)) - 3|) + 2 and g(x) = 3|(5(-2cos(1πx - 2) - 3)^2 + 2(-2cos(1πx - 2) - 3) - 3) + 2| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((2(1|(1x - 4) - 2| - 1))/(3(1|(1x - 4) - 2| - 1) - 1))^2 + 3 and g(x) = 5(-2cos(1π(-1|x - 2|) + 1)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation -3|(5((-3(-1x^2 + 2x + 1) + 1)/(-3(-1x^2 + 2x + 1) - 3)) + 5) + 3| - 2 = -2|(-5(2x - 1)^2 - 3(2x - 1) + 4) - 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation (2(-3(2cos(1πx + 1) + 2) + 5) + 1)/(-3(-3(2cos(1πx + 1) + 2) + 5) + 1) = 2cos(1π((2x - 2)/(-3x + 4)) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "How many solutions does the equation 1|(3(3cos(2π((3x - 1)/(-1x - 2)) + 1) + 1)^2 + 5(3cos(2π((3x - 1)/(-1x - 2)) + 1) + 1)) + 1| + 3 = -1cos(2π(-1(1x + 3)^2 + 4(1x + 3)) - 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 117 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((2(2sin(1π(-1x^2 + 5x - 3) + 3)) + 2)/(2(2sin(1π(-1x^2 + 5x - 3) + 3)) - 3)) + 1| - 3 and g(x) = -2|(-4(-3cos(2πx - 3) + 3)^2 - 2) + 2| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 63 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((1(-3sin(1πx + 3) + 3) + 3)/(-1(-3sin(1πx + 3) + 3) - 1)) + 1| - 3 and g(x) = -1cos(1π((-2x + 3)/(-1x - 2)) + 2) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 24 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(1e^(1(-1x - 1)) - 2) + 1)/(1(1e^(1(-1x - 1)) - 2) + 3) and g(x) = (-2(3sin(2πx - 2) + 1) + 3)/(-2(3sin(2πx - 2) + 1) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 39 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(2π(-4(-1|x - 2| + 2) - 4) + 3) + 2 and g(x) = 2|(-1e^(-1x) - 1) + 1| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(2π(1|((3x - 3)/(1x + 1)) - 3|) + 2) - 1 and g(x) = 1e^(-1(-3x - 5)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "How many solutions does the equation 2cos(2π(2|(-1x - 5) - 3| + 2) - 2) - 3 = 1(1cos(3πx + 2) + 1)^2 + 5(1cos(3πx + 2) + 1) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(2sin(2π(3|(-5x^2 - 3x + 1) - 3| + 2) + 3) + 2) + 4 and g(x) = -1(1((3x + 2)/(-1x - 2))^2 + 5((3x + 2)/(-1x - 2)) - 4) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 84 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((-1(-2|(3sin(3πx + 2) + 3) + 1| - 2) + 1)/(3(-2|(3sin(3πx + 2) + 3) + 1| - 2) + 3))^2 + 1((-1(-2|(3sin(3πx + 2) + 3) + 1| - 2) + 1)/(3(-2|(3sin(3πx + 2) + 3) + 1| - 2) + 3)) - 4 and g(x) = -1|((1(1x - 3) - 3)/(-2(1x - 3) - 3)) +... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(2(-2sin(3π((-2x + 1)/(-3x + 4)) - 2) - 3)^2 - 1(-2sin(3π((-2x + 1)/(-3x + 4)) - 2) - 3) - 2) + 4 and g(x) = 4(-2(2cos(1πx + 2) + 2) + 4)^2 - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 63 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(3π(2(3x + 3)^2 + 5(3x + 3) + 5) + 2) and g(x) = (1(-1x^2 + 4x + 2) + 1)/(3(-1x^2 + 4x + 2) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1471 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|((-2(-2x^2 - 3) + 2)/(-1(-2x^2 - 3) - 1)) + 1| - 2 and g(x) = -3cos(1π(3x) + 3) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 54 | MATH |
[
{
"content": "How many solutions does the equation -2sin(1π(2|(5((-2x + 1)/(3x + 3))^2 + 1((-2x + 1)/(3x + 3)) - 1) + 3| - 3) - 3) + 1 = (1(-1(1x^2 - 4x + 3) - 2) + 3)/(1(-1(1x^2 - 4x + 3) - 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 155 | MATH |
[
{
"content": "How many solutions does the equation 2|(2(-2sin(2πx) + 2)^2 - 4(-2sin(2πx) + 2) - 4) - 2| - 2 = 3|(2cos(1πx - 3) + 2) + 2| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(3π(-1(-3|((-3x - 3)/(-3x + 2)) + 1| + 3)^2) + 1) and g(x) = 2sin(2π((2(-4x^2 + 4x + 5) + 2)/(3(-4x^2 + 4x + 5) + 1)) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-1|(2cos(2πx - 1) + 1) + 1| + 3) + 1)/(1(-1|(2cos(2πx - 1) + 1) + 1| + 3)) and g(x) = 3(-2|x| + 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "How many solutions does the equation 3(4(-3sin(2πx - 2) + 3) - 1)^2 + 4(4(-3sin(2πx - 2) + 3) - 1) + 3 = 2sin(1π(1x^2 - 1x + 5) + 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 46 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-3(1((2x + 2)/(1x + 2))^2 - 4((2x + 2)/(1x + 2)) - 4) + 5) - 1| + 3 and g(x) = (-2(-1sin(3π(1x^2 - 4)) + 2) + 1)/(2(-1sin(3π(1x^2 - 4)) + 2) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 47 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(4(2sin(3πx) + 1) - 3)^2 - 2(4(2sin(3πx) + 1) - 3) - 3 and g(x) = (3(1cos(3πx - 1) - 1) - 1)/(-2(1cos(3πx - 1) - 1) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1((2(-2sin(1πx - 1) + 1) - 1)/(1(-2sin(1πx - 1) + 1) + 1)) and g(x) = 3|(3cos(3πx - 3) - 3) + 2| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(2((-2(3cos(3πx + 3) - 3) + 1)/(-1(3cos(3πx + 3) - 3) + 2)) - 1) - 3| - 2 and g(x) = (3(-2|(5x + 5) + 1|) + 3)/(-1(-2|(5x + 5) + 1|) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 58 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3e^(2(-2|((-1x - 2)/(-1x)) - 2|)) - 1 and g(x) = -2sin(2π((2x + 1)/(3x + 1)) - 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|((1(-1(-2x + 2)^2 - 5(-2x + 2) - 5))/(-1(-1(-2x + 2)^2 - 5(-2x + 2) - 5) - 2)) + 3| - 3 and g(x) = -3|(4((2x - 2)/(-2x - 3))^2 - 4((2x - 2)/(-2x - 3)) - 3) - 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(2π(-1|(-1(2x^2 + 2) + 4) + 2| - 3) + 1) + 3 and g(x) = -3(1((3x - 2)/(-1x + 2))^2 + 4((3x - 2)/(-1x + 2)) + 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 124 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4((3(1(3|x| + 3)^2 - 4(3|x| + 3) + 5) + 3)/(3(1(3|x| + 3)^2 - 4(3|x| + 3) + 5) + 2)) and g(x) = (-3(-3(2cos(3πx) - 2)^2 - 1(2cos(3πx) - 2) + 4) + 3)/(-1(-3(2cos(3πx) - 2)^2 - 1(2cos(3πx) - 2) + 4) - 3) intersect in the interval [-10, 10].\n\nPresent t... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(1(-2sin(2π(-1x^2 - 2x + 5) + 2) + 1) + 5))/(2(1(-2sin(2π(-1x^2 - 2x + 5) + 2) + 1) + 5) - 1) and g(x) = -1|(1sin(1π(-3x^2 - 3x - 2) + 3) + 2) + 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{You... | 602 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(3π((-3(-2x^2 + 2x + 1) - 3)/(3(-2x^2 + 2x + 1))) - 3) - 1 and g(x) = -1cos(3π((3x + 1)/(2x + 3)) - 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "How many solutions does the equation 1((-2(3|x| - 2) + 1)/(3(3|x| - 2) + 2))^2 - 2((-2(3|x| - 2) + 1)/(3(3|x| - 2) + 2)) - 4 = 4(1x^2 + 4x + 4) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(1|(2sin(1π(-2x^2 - 3x + 3) - 1) - 2) + 1| - 3) + 4 and g(x) = 2e^(-2(-1(2x + 5)^2 - 5(2x + 5) + 2)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 68 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-5(-2cos(3πx) + 3)^2 + 5(-2cos(3πx) + 3) + 3) - 3| - 3 and g(x) = -1|(-2cos(3πx) + 2) + 2| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(2π(-2|(5x^2 + 4x + 1) + 1|) + 3) + 3 and g(x) = (-3(-1x - 5))/(-3(-1x - 5) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 522 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|((-1(-3cos(3π(-1x) - 3) + 1) + 3)/(3(-3cos(3π(-1x) - 3) + 1) - 2)) + 3| - 3 and g(x) = 4((-2(5x - 4) - 2)/(1(5x - 4) - 2))^2 - 1((-2(5x - 4) - 2)/(1(5x - 4) - 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \... | 107 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2((-2(-2|(3x - 3) - 2| - 3) - 2)/(-2(-2|(3x - 3) - 2| - 3) + 3))^2 + 1((-2(-2|(3x - 3) - 2| - 3) - 2)/(-2(-2|(3x - 3) - 2| - 3) + 3)) - 2 and g(x) = (-1(-1sin(2π(1x) - 2) - 3))/(-2(-1sin(2π(1x) - 2) - 3) - 2) where -10 ≤ x ≤ ... | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-1(5x - 4)^2 - 5(5x - 4) + 1)| + 2 and g(x) = -5(-2sin(3πx + 2) + 3) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(1π((3(4x^2 + 2x - 4) + 2)/(1(4x^2 + 2x - 4) + 3)) + 2) and g(x) = (-2(5x^2 - 1x + 3) + 2)/(-2(5x^2 - 1x + 3) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 87 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2(1cos(3π(-2|x - 3|) + 2))^2 + 4(1cos(3π(-2|x - 3|) + 2)) - 3) - 2)/(-1(2(1cos(3π(-2|x - 3|) + 2))^2 + 4(1cos(3π(-2|x - 3|) + 2)) - 3) - 2) and g(x) = 2|((-1(-3sin(3πx + 3) + 3) - 1)/(-2(-3sin(3πx + 3) + 3) + 3)) - 1| in the ran... | 114 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(3sin(2π(-1x - 2) - 3))^2 + 1 and g(x) = (-2(4x + 4) + 2)/(-2(4x + 4) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 43 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(3(2e^(-1(-4x - 3)) - 2)^2 + 3(2e^(-1(-4x - 3)) - 2) - 3) + 1)/(2(3(2e^(-1(-4x - 3)) - 2)^2 + 3(2e^(-1(-4x - 3)) - 2) - 3)) and g(x) = -2cos(2π(-3|(3x^2 - 5) - 1| - 1) + 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in L... | 1656 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(3|(5((-2x + 1)/(-1x + 3))^2 + 3((-2x + 1)/(-1x + 3)) + 1) - 3| - 2) + 3 and g(x) = (-2(3sin(1π(-1x^2 + 3) - 3) + 1))/(1(3sin(1π(-1x^2 + 3) - 3) + 1) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Y... | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(2cos(1π((2x + 1)/(-1x + 2)) + 2) + 3) + 2 and g(x) = 5((2x - 2)/(-1x - 2))^2 - 5((2x - 2)/(-1x - 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-1|(-1(1x + 4)^2 - 5(1x + 4) - 3)| + 3) + 1)/(-2(-1|(-1(1x + 4)^2 - 5(1x + 4) - 3)| + 3)) and g(x) = (3(3(2x^2 - 3x - 2) - 5) - 2)/(-2(3(2x^2 - 3x - 2) - 5) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Yo... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(2(-3sin(3π((2x - 3)/(-1x - 3)) - 1) - 2) - 4) + 1| - 1 and g(x) = -3e^(1(-3((-1x - 2)/(3x + 2))^2 - 2((-1x - 2)/(3x + 2)) - 2)) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 93 | MATH |
[
{
"content": "How many solutions does the equation -1cos(1π(-3(3|x - 3|)^2 - 3(3|x - 3|) - 4) - 3) - 2 = -3e^(-1((2x - 1)/(1x + 3))) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2e^(-1(2|((-2x + 3)/(-2x - 3)) + 1| - 2)) + 1 and g(x) = 3sin(2π(-2|x - 1| + 2) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 76 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2((2(2x^2 + 3x - 1) - 1)/(1(2x^2 + 3x - 1))) + 5 and g(x) = 2cos(2π(4x - 4) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 152 | MATH |
[
{
"content": "How many solutions does the equation 2(-1|(5((-1x - 2)/(-3x - 2))^2 - 3((-1x - 2)/(-3x - 2)) + 4) - 1| + 3) + 2 = 5(-3cos(3π(-2|x + 1| - 2)) + 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 114 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-4(-3(3|x - 1| - 2)^2 - 1(3|x - 1| - 2) + 4)) + 1)/(3(-4(-3(3|x - 1| - 2)^2 - 1(3|x - 1| - 2) + 4)) - 2) and g(x) = 1|(2((1x - 3)/(2x))^2 + 3((1x - 3)/(2x)) - 2) - 2| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex ... | 6 | MATH |
[
{
"content": "How many solutions does the equation 2cos(2π(5(3|x + 3| + 1) + 5) + 3) + 2 = -1((1x - 1)/(1x - 2)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many solutions does the equation (2(5(2sin(2π(-2|x + 1| + 1) - 1) - 1)^2 + 4(2sin(2π(-2|x + 1| + 1) - 1) - 1) - 2) + 3)/(2(5(2sin(2π(-2|x + 1| + 1) - 1) - 1)^2 + 4(2sin(2π(-2|x + 1| + 1) - 1) - 1) - 2) + 2) = 4((-3(1sin(2πx - 3) - 1))/(3(1sin(2πx - 3) - 1) - 2))^2 - 4((-3(1sin(2πx - 3) - 1))/(... | 98 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(2sin(3π(-1x - 1) - 1) + 1) + 3| - 2 and g(x) = -2|(-2cos(3πx + 2) + 2) + 3| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many solutions does the equation -3((-1(-2e^(-1(1|x + 3| - 1)) - 1) - 1)/(-1(-2e^(-1(1|x + 3| - 1)) - 1) - 2)) - 1 = 1|(3cos(2π(1x - 3) + 1) + 3) - 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((3(-3(-3sin(3πx + 2) - 1) + 2) - 3)/(2(-3(-3sin(3πx + 2) - 1) + 2))) + 3| + 2 and g(x) = -2(3e^(2(-2x + 3)))^2 + 4(3e^(2(-2x + 3))) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 50 | MATH |
[
{
"content": "How many solutions does the equation -3sin(2π(-3|(5x^2 - 5x)| - 2) - 1) + 3 = -5(2|x + 3| - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 92 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-1cos(2π(-4x - 1) + 3) + 1)^2 - 4(-1cos(2π(-4x - 1) + 3) + 1) + 2 and g(x) = -2|(-3x - 5) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|((-1(-1x^2 - 5x - 1) - 2)/(-2(-1x^2 - 5x - 1) - 3)) - 3| + 2 and g(x) = 3cos(2π(2x^2 + 2x - 5) + 3) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 748 | MATH |
No dataset card yet
- Downloads last month
- 7