messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-4(3cos(2πx + 2) - 2)^2 + 1(3cos(2πx + 2) - 2) - 3) + 1 and g(x) = 4(5x + 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-1cos(2π(1x^2 - 3x - 1) + 2) + 3)^2 - 5(-1cos(2π(1x^2 - 3x - 1) + 2) + 3) - 1 and g(x) = 2|(-3x + 1) - 1| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "How many solutions does the equation 1cos(1π(-1(4x - 1)^2 + 4(4x - 1) + 3) + 2) + 3 = 2|(-2cos(2πx - 1) + 2) - 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 381 | MATH |
[
{
"content": "How many solutions does the equation -3(-1((-1x + 1)/(-1x + 2)) - 1)^2 + 4(-1((-1x + 1)/(-1x + 2)) - 1) + 5 = -3(2x^2 + 5x - 1)^2 - 3(2x^2 + 5x - 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": "How many points of intersection exist between the functions f(x) = (-2((-2(1x^2 + 2x + 5) + 3)/(1(1x^2 + 2x + 5))))/(-2((-2(1x^2 + 2x + 5) + 3)/(1(1x^2 + 2x + 5))) + 3) and g(x) = -2cos(1π(-1x - 2) - 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((1(-3sin(1πx - 2) - 2) + 3)/(1(-3sin(1πx - 2) - 2))) - 4 and g(x) = -2cos(2π(-2x^2 + 2x - 4) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 385 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(2π((1(-1x^2 + 3x) - 3)/(-1(-1x^2 + 3x) + 2)) + 3) + 2 and g(x) = (-1(3|x + 2| + 1) + 1)/(-2(3|x + 2| + 1) + 1) in the range -10 ≤ x ≤ 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) = 5(1((-2x - 1)/(-1x)) - 2)^2 - 1(1((-2x - 1)/(-1x)) - 2) - 1 and g(x) = 2sin(1π(5x^2 + 1x + 1) + 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 433 | MATH |
[
{
"content": "How many solutions does the equation 1((-3(-3cos(3πx + 3)) - 3)/(-2(-3cos(3πx + 3)) - 1))^2 - 3((-3(-3cos(3πx + 3)) - 3)/(-2(-3cos(3πx + 3)) - 1)) - 1 = 5(1e^(1x) + 2)^2 - 2(1e^(1x) + 2) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 70 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-3sin(2π(-2|x - 3| - 2) + 2) + 1) + 2)/(-1(-3sin(2π(-2|x - 3| - 2) + 2) + 1)) and g(x) = 1sin(3π(3|x - 1| - 1) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 118 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-1(3cos(1πx - 1) - 1)^2 - 4(3cos(1πx - 1) - 1) + 1) + 2)/(-1(-1(3cos(1πx - 1) - 1)^2 - 4(3cos(1πx - 1) - 1) + 1) - 2) and g(x) = -3(-2e^(1x) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(3sin(1π(-2|x - 2|) - 3) + 1) + 5 and g(x) = (-1((1x)/(3x - 2)) - 2)/(1((1x)/(3x - 2)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "How many solutions does the equation 3|((2((1x + 2)/(-2x + 1)) + 3)/(-3((1x + 2)/(-2x + 1)) + 2))| + 2 = -3sin(1π(-5x^2 - 3x + 5) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 99 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(3π((3(3x^2 - 1x - 3) - 1)/(-1(3x^2 - 1x - 3))) - 2) and g(x) = 2sin(3π(3|x + 3| - 2) - 1) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 197 | MATH |
[
{
"content": "How many solutions does the equation 4(2(-1|x - 2| + 3) + 2)^2 + 3(2(-1|x - 2| + 3) + 2) - 5 = -3((2x - 3)/(-2x + 1)) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(3cos(3π(2x^2 - 1x) - 1) - 1) - 1 and g(x) = 5(-1|x| + 2)^2 - 1(-1|x| + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 129 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3((3(3|x + 1| + 3) - 2)/(3(3|x + 1| + 3))) - 3)/(-1((3(3|x + 1| + 3) - 2)/(3(3|x + 1| + 3))) - 1) and g(x) = 1(-3cos(3πx - 3) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(1π(2(2x^2)) - 2) - 1 and g(x) = 2sin(3π(1|x + 1| + 1) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 786 | MATH |
[
{
"content": "How many solutions does the equation (-1(1|(3sin(2πx) + 1) - 1| - 2) + 3)/(1(1|(3sin(2πx) + 1) - 1| - 2) + 1) = 1|(-1x^2 - 1) + 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 68 | MATH |
[
{
"content": "How many solutions does the equation -1|(-3cos(3π(-4x^2 + 5x) - 1) + 1) - 2| - 3 = (-2(-3x + 1) + 2)/(1(-3x + 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2e^(1(-3(1|x - 3| + 3)^2 - 3(1|x - 3| + 3) - 5)) + 2 and g(x) = 2cos(3π(-4x^2 + 1x + 4)) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1432 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((-1(-2x^2 + 5x + 4) + 1)/(-2(-2x^2 + 5x + 4) - 1))^2 + 5((-1(-2x^2 + 5x + 4) + 1)/(-2(-2x^2 + 5x + 4) - 1)) - 3 and g(x) = -3(2x^2 - 3x + 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(2π(-1|(-2|x + 2| + 3) - 3|) - 3) and g(x) = 1e^(2((2x + 2)/(-1x + 2))) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 67 | MATH |
[
{
"content": "How many solutions does the equation 1(2cos(2π(2x^2 - 2x - 1)))^2 - 4(2cos(2π(2x^2 - 2x - 1))) + 2 = 1cos(2π(1x + 4) + 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 802 | MATH |
[
{
"content": "How many solutions does the equation -2cos(2π(-2|(5x)| + 3) + 3) = 2sin(2π(1|x + 2| + 1)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 200 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-2sin(1π(2|x + 3| - 3) + 3)) - 4 and g(x) = -3(1sin(2πx) + 2)^2 + 2(1sin(2πx) + 2) + 4 in the range -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) = -1|((-1(-3sin(3πx - 3) - 2) + 3)/(-1(-3sin(3πx - 3) - 2) - 2))| - 2 and g(x) = -2(-2cos(2πx + 1) - 3)^2 - 3(-2cos(2πx + 1) - 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 100 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2|(-3e^(-1x) - 3) + 2| - 3)^2 - 5(2|(-3e^(-1x) - 3) + 2| - 3) + 4 and g(x) = -3cos(3π(-2x + 5) + 1) - 3 in the range -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) = (3((-1(1x^2 - 1x - 3) + 2)/(-3(1x^2 - 1x - 3))) - 1)/(-2((-1(1x^2 - 1x - 3) + 2)/(-3(1x^2 - 1x - 3))) - 3) and g(x) = (-2(3x^2 - 2x + 1) + 3)/(2(3x^2 - 2x + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{You... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(5(-3x^2 + 4x - 1)^2 + 1(-3x^2 + 4x - 1) - 3)^2 + 5(5(-3x^2 + 4x - 1)^2 + 1(-3x^2 + 4x - 1) - 3) - 2 and g(x) = 1((2x + 3)/(3x - 2)) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 5 | MATH |
[
{
"content": "How many solutions does the equation -3sin(3π(-2((-3x - 2)/(1x + 3))^2 - 4((-3x - 2)/(1x + 3)) - 4) - 3) + 2 = (3(-2|x - 1| + 2))/(2(-2|x - 1| + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 448 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-2cos(1π(2|x - 1| + 1)) + 1) - 1)/(1(-2cos(1π(2|x - 1| + 1)) + 1) - 3) and g(x) = (2((1x - 1)/(-3x + 1)) + 1)/(2((1x - 1)/(-3x + 1)) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 27 | MATH |
[
{
"content": "How many solutions does the equation -2(-5(-1x^2 - 3x) + 2)^2 - 2(-5(-1x^2 - 3x) + 2) + 5 = -3(3cos(3πx) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-3(-1e^(-1x) + 1)) - 4 and g(x) = 1(-1cos(1πx - 1) + 3)^2 + 3(-1cos(1πx - 1) + 3) - 5 where -10 ≤ x ≤ 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) = 4((2(1x^2 + 1x - 4) - 1)/(-2(1x^2 + 1x - 4))) + 5 and g(x) = -3|(-3x - 4)| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(2sin(3π(-1|x|) - 2) - 3)^2 - 2 and g(x) = -5(-3cos(2πx) + 2) - 1 in the range -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) = -2cos(1π(5((-1x - 3)/(2x + 2))^2 - 5((-1x - 3)/(2x + 2)) + 2) - 3) - 3 and g(x) = -2|(-1cos(1πx + 2) + 1) - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 47 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(1(2|x + 1| - 2)^2 - 1(2|x + 1| - 2) - 2) + 5 and g(x) = -1sin(1π(-5x - 5) - 2) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(2cos(2π(1x) - 3) + 2) - 1)/(-1(2cos(2π(1x) - 3) + 2)) and g(x) = 4(-1|x| + 1) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(2|(-4x - 2) - 3| + 2) - 3)/(2(2|(-4x - 2) - 3| + 2) + 2) and g(x) = 1(4x^2 + 2x - 5)^2 + 3(4x^2 + 2x - 5) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(-5(1x^2 - 5x + 1))| - 3 and g(x) = (3(2e^(2x) + 3) - 2)/(2(2e^(2x) + 3) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(4(1x^2 - 1x - 1) + 1) - 2| - 2 and g(x) = (3(-2e^(1x) - 2))/(1(-2e^(1x) - 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(2π((-3(3x + 1) - 3)/(-2(3x + 1) + 3))) + 1 and g(x) = 1(-2x^2 - 1x - 5)^2 + 5(-2x^2 - 1x - 5) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-2sin(1π(3x^2 + 4x - 5) + 1) + 2) + 1| + 3 and g(x) = 1(-4x - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((1(-2|x + 2| + 3) + 2)/(1(-2|x + 2| + 3) + 1)) - 2 and g(x) = -3(-3cos(1πx - 3) + 2)^2 - 4(-3cos(1πx - 3) + 2) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many solutions does the equation 1|(3sin(1π((-2x + 3)/(-3x - 1)) + 3) - 1) - 3| - 3 = -2cos(1π(-3x^2 - 5x + 2) - 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 584 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1sin(2π(4(3|x + 1| - 3)^2 + 1(3|x + 1| - 3) + 1) - 3) - 1 and g(x) = (3(-5x^2 - 2x + 1) + 2)/(3(-5x^2 - 2x + 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2cos(2π(-1|x|) - 3) + 1)^2 + 3 and g(x) = -1((3x + 3)/(-2x - 1))^2 - 1((3x + 3)/(-2x - 1)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 62 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(2π(5((-3x - 3)/(1x - 2))^2 + 3((-3x - 3)/(1x - 2)))) - 1 and g(x) = (-1(1|x + 1| - 1) - 3)/(3(1|x + 1| - 1) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 616 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(1π((-3(-1|x| - 2) - 3)/(-1(-1|x| - 2) - 3))) + 1 and g(x) = -2cos(1π((1x - 2)/(-3x - 3)) - 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 133 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(1π((1(2x^2 - 4x - 1) - 1)/(1(2x^2 - 4x - 1) + 3)) + 3) + 1 and g(x) = -1(-4x^2 + 2x + 4) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-3(4x) + 2)/(-1(4x) + 2)) - 5 and g(x) = -3|(-3|x + 2| + 3) - 1| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many solutions does the equation -3(3e^(1(-1x^2 - 4x - 1)) - 3)^2 + 4(3e^(1(-1x^2 - 4x - 1)) - 3) - 2 = 1sin(3π((-1x - 1)/(3x + 3)) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-4((3x - 3)/(-1x))^2 + 5((3x - 3)/(-1x))) + 2)/(-3(-4((3x - 3)/(-1x))^2 + 5((3x - 3)/(-1x))) - 2) and g(x) = -3sin(2π(-4x + 2) + 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 162 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(3π(-3(-3|x + 1| - 2) + 5) + 3) - 3 and g(x) = 2cos(3π((2x + 2)/(-2x - 2)) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 540 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(3(1e^(-1x)) - 2)^2 + 1(3(1e^(-1x)) - 2) - 2 and g(x) = -2|((3x + 1)/(-1x - 1))| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(2sin(1π(-2x^2 - 2x - 5) + 2)) - 2)/(-1(2sin(1π(-2x^2 - 2x - 5) + 2)) - 3) and g(x) = 2e^(-2(2|x + 3| + 3)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 402 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(5(-2|x + 2| + 2)^2 - 4(-2|x + 2| + 2))^2 + 1(5(-2|x + 2| + 2)^2 - 4(-2|x + 2| + 2)) + 5 and g(x) = 5(2|x + 2| + 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(3π(1((-2x - 1)/(-1x - 2)) - 5)) - 1 and g(x) = -1|(-1|x - 2| + 3) - 3| where -10 ≤ x ≤ 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) = -1(1|(-2cos(1πx - 1) - 2) + 3| - 3)^2 + 4(1|(-2cos(1πx - 1) - 2) + 3| - 3) + 3 and g(x) = (2(2sin(2πx + 1) - 2) + 3)/(2(2sin(2πx + 1) - 2) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 57 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(1π(-3|(-3|x + 2|)| - 2) - 1) + 2 and g(x) = (1(-2|x - 3| + 1) + 2)/(1(-2|x - 3| + 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 152 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(3π(-5(3x^2 + 5x) + 5) - 3) + 2 and g(x) = 3|(-2cos(3πx + 2) - 1) + 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 684 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-1e^(-2(1x^2 + 5)) - 1) - 1 and g(x) = -3(1cos(1πx - 2)) - 1 intersect 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 3((3((-2x - 3)/(-2x)) + 2)/(-2((-2x - 3)/(-2x)) - 1)) + 1 = (3(3sin(2πx - 3)) - 3)/(3(3sin(2πx - 3))) 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) = 1(2e^(-1(-1|x + 2|)) - 3) - 5 and g(x) = 2(3sin(2πx) - 3)^2 + 3(3sin(2πx) - 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation -3sin(3π(-4(-5x + 4)^2 + 1) - 2) + 3 = 2(-1sin(2πx + 2) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1421 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-4(-1sin(2πx) - 3)^2 - 1(-1sin(2πx) - 3) - 5)^2 - 3(-4(-1sin(2πx) - 3)^2 - 1(-1sin(2πx) - 3) - 5) + 3 and g(x) = -5(3|x + 1| + 2)^2 - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 28 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-3(-2x - 4) - 2)^2 and g(x) = 2cos(3π((-2x)/(2x + 3)) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(1|(-3|x + 2| - 1) + 3| + 3) + 2 and g(x) = (1(-2cos(2πx + 3) + 2) + 2)/(1(-2cos(2πx + 3) + 2) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(2π(-5(1|x - 2| - 3)^2 - 4(1|x - 2| - 3) + 5) + 2) + 2 and g(x) = 1e^(2(3x)) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 695 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(2π(-1|(-3x - 2) - 3| - 2)) + 1 and g(x) = -2cos(2π(1|x| - 2) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 122 | MATH |
[
{
"content": "How many solutions does the equation 3|(1cos(1π((-2x + 2)/(1x + 1)) + 1) + 3) - 3| - 2 = -3(3x + 5)^2 + 1(3x + 5) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-4(-1e^(1x) + 2) - 2) + 1)/(2(-4(-1e^(1x) + 2) - 2) - 3) and g(x) = -1(1sin(2πx + 3) - 3)^2 - 2(1sin(2πx + 3) - 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 39 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(3π(2(2x + 5)^2 + 1(2x + 5) - 5) + 2) + 1 and g(x) = 1cos(2π(1x^2 + 3x + 5) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(2π((-1(-3x) + 2)/(-2(-3x) - 2)) - 3) and g(x) = 2cos(1π(-2|x - 3| - 2) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(4(4x^2 - 5x - 3)^2 - 5)^2 + 1(4(4x^2 - 5x - 3)^2 - 5) - 1 and g(x) = -1|(-1sin(2πx - 3) - 2) - 2| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(3sin(3π(4x^2 - 2x + 5)) + 2)^2 - 1(3sin(3π(4x^2 - 2x + 5)) + 2) + 3 and g(x) = 3|(-1|x - 1| - 1) - 2| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1511 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5((3(2cos(3πx + 1)) + 3)/(-3(2cos(3πx + 1)) + 2)) + 2 and g(x) = 2cos(2π(-2x^2 + 3x + 1) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 309 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1cos(1π(-2|x - 3|) - 3) + 1)^2 - 3(1cos(1π(-2|x - 3|) - 3) + 1) + 5 and g(x) = 2cos(3π((-3x + 3)/(-1x - 2)) + 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 86 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3((-1(5x^2 - 4x - 1) - 1)/(3(5x^2 - 4x - 1) + 1))^2 + 4((-1(5x^2 - 4x - 1) - 1)/(3(5x^2 - 4x - 1) + 1)) + 1 and g(x) = -2((-2x)/(3x + 2))^2 + 1((-2x)/(3x + 2)) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-3(-1e^(-2x) + 1) + 2) - 2| and g(x) = (-1(2sin(2πx - 1) + 2) + 1)/(2(2sin(2πx - 1) + 2) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(3π(-3|(-1x + 4) - 1| + 3) + 2) and g(x) = (-3(-4x + 1) + 1)/(3(-4x + 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 181 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2((-2(-3cos(1πx + 1) - 2) - 2)/(-3(-3cos(1πx + 1) - 2) - 1))^2 + 5((-2(-3cos(1πx + 1) - 2) - 2)/(-3(-3cos(1πx + 1) - 2) - 1)) + 4 and g(x) = -1((2x - 3)/(-2x)) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{You... | 39 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1(2e^(-2x) - 3)^2 - 2(2e^(-2x) - 3) + 2) + 5 and g(x) = 2sin(1π(-5x + 4) - 2) + 2 where -10 ≤ x ≤ 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) = -3e^(-1(5(1x - 5)^2 - 3(1x - 5) - 3)) - 2 and g(x) = -4(2e^(-1x) - 3)^2 + 4(2e^(-1x) - 3) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2((-3(-3sin(1πx - 2) - 3) + 1)/(2(-3sin(1πx - 2) - 3) - 3))^2 - 3((-3(-3sin(1πx - 2) - 3) + 1)/(2(-3sin(1πx - 2) - 3) - 3)) and g(x) = 2|(-2x + 4) - 2| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Y... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(2π(-2|(1|x + 3|) + 2| - 3) - 3) - 3 and g(x) = 1((1x - 1)/(1x - 2)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 46 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3sin(1π((-2((1x - 3)/(-1x + 2)) + 3)/(2((1x - 3)/(-1x + 2)) + 2)) + 1) and g(x) = -1|(4x + 5)| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-1e^(2(1x^2 - 3x - 3)) + 3)^2 + 5(-1e^(2(1x^2 - 3x - 3)) + 3) + 1 and g(x) = -2((-3x + 1)/(1x + 3)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(1π((1((-1x + 1)/(1x - 3)) + 3)/(3((-1x + 1)/(1x - 3)) + 2)) - 1) - 1 and g(x) = (-3((-2x - 1)/(-3x + 1)))/(-1((-2x - 1)/(-3x + 1)) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 134 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(4(-1x^2 + 3x + 1)^2 - 1(-1x^2 + 3x + 1) - 4) + 1| + 1 and g(x) = -3|(5x^2 - 5x - 4) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|((1(-2cos(1πx - 1)) + 3)/(2(-2cos(1πx - 1)) + 3)) - 1| and g(x) = 3|(1x^2 - 5x)| - 1 intersect in the interval [-10, 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) = 4(1|(3cos(2πx) + 3) - 1| - 3) + 2 and g(x) = 3(-2x - 2)^2 + 4(-2x - 2) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation -2|(-2|(3sin(2πx) + 3) - 3| + 2) - 3| = 5(-3|x + 3|) 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) = -2sin(2π(-4(-3x - 3) - 3) - 2) + 3 and g(x) = -3(2|x + 1|)^2 + 4 in the range -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) = -3sin(1π((-2((-3x + 3)/(-3x - 3)) - 2)/(-2((-3x + 3)/(-3x - 3)) - 3)) + 2) + 1 and g(x) = (-3(2|x + 3| - 3) + 1)/(3(2|x + 3| - 3) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-2cos(2π(1x^2 + 1x + 3) + 1) - 2)^2 + 3(-2cos(2π(1x^2 + 1x + 3) + 1) - 2) + 4 and g(x) = (1(2|x - 3| - 3) - 2)/(1(2|x - 3| - 3)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 62 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-1cos(1π(-4x^2 - 4x + 4) + 1) - 2))/(3(-1cos(1π(-4x^2 - 4x + 4) + 1) - 2) - 2) and g(x) = 2cos(1π(1x - 4) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 52 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-3(-4x^2 + 4x + 3)^2 + 2(-4x^2 + 4x + 3) - 2)^2 + 3(-3(-4x^2 + 4x + 3)^2 + 2(-4x^2 + 4x + 3) - 2) - 4 and g(x) = -3e^(-1(-3|x + 1| - 1)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 5 | MATH |
[
{
"content": "How many solutions does the equation (-3(3sin(3π(-3x^2 + 2x) - 1) + 3) - 3)/(-2(3sin(3π(-3x^2 + 2x) - 1) + 3) + 3) = -2((-3x - 1)/(-3x - 1))^2 - 3((-3x - 1)/(-3x - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 428 | MATH |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.