messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = -4(-4(3cos(1π(4x - 1) + 2) + 2)^2 - 1(3cos(1π(4x - 1) + 2) + 2) - 1)^2 - 3(-4(3cos(1π(4x - 1) + 2) + 2)^2 - 1(3cos(1π(4x - 1) + 2) + 2) - 1) - 1 and g(x) = -5((3(4x^2 - 3x - 1) - 1)/(2(4x^2 - 3x - 1) + 2)) + 3 intersect in the interval [-10, 10].\n\nPr... | 149 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1sin(1π((-3(5(-5x^2 + 5x - 1) - 2) - 3)/(3(5(-5x^2 + 5x - 1) - 2) - 1)) - 2) - 3 and g(x) = -3|(1(1|x + 1| - 3) + 1) - 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 4 | MATH |
[
{
"content": "How many solutions does the equation 4(-1sin(1π(2(-4x^2 + 1x - 3) - 4) + 3) - 1) - 2 = -1(2|(1x - 4) - 3|) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 90 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(2(3|(-2x - 4) - 2| - 3) - 2)^2 - 4(2(3|(-2x - 4) - 2| - 3) - 2) - 3 and g(x) = 2|(-1sin(2π(-1x^2 + 3x + 5) + 3) + 2) + 2| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-5(-1(5x^2 + 4x - 3)^2 - 2(5x^2 + 4x - 3) + 1)^2 - 4(-1(5x^2 + 4x - 3)^2 - 2(5x^2 + 4x - 3) + 1) + 5) + 4 and g(x) = 3(3sin(3π(-1x^2 - 5x - 5) + 1) + 2)^2 - 4(3sin(3π(-1x^2 - 5x - 5) + 1) + 2) - 3 where -10 ≤ x ≤ 10.\n\nPr... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((1((-3(2|x + 2| - 3))/(3(2|x + 2| - 3) - 2)) + 3)/(-1((-3(2|x + 2| - 3))/(3(2|x + 2| - 3) - 2)) + 2))^2 + 5((1((-3(2|x + 2| - 3))/(3(2|x + 2| - 3) - 2)) + 3)/(-1((-3(2|x + 2| - 3))/(3(2|x + 2| - 3) - 2)) + 2)) - 3 and g(x) = -2|... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-1(-1(2|x + 1| - 1)^2 + 4(2|x + 1| - 1) - 5)^2 - 2(-1(2|x + 1| - 1)^2 + 4(2|x + 1| - 1) - 5) + 4) - 2| + 2 and g(x) = 2((-2(1x + 5) - 2)/(2(1x + 5) - 1))^2 + 2((-2(1x + 5) - 2)/(2(1x + 5) - 1)) - 3 where -10 ≤ x ≤ 10.\n\n... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3((1(-3sin(3π(-2|x - 2|) + 3) - 2) + 2)/(-1(-3sin(3π(-2|x - 2|) + 3) - 2) + 3)))/(-1((1(-3sin(3π(-2|x - 2|) + 3) - 2) + 2)/(-1(-3sin(3π(-2|x - 2|) + 3) - 2) + 3)) + 2) and g(x) = 3|((1(-4x + 3) - 1)/(2(-4x + 3) - 1)) - 1| + 1 intersect in the interval... | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1(1((3x - 2)/(-3x + 4)) + 2)^2 - 5(1((3x - 2)/(-3x + 4)) + 2) + 3))/(-3(1(1((3x - 2)/(-3x + 4)) + 2)^2 - 5(1((3x - 2)/(-3x + 4)) + 2) + 3) - 3) and g(x) = (-2(5(1cos(3πx) + 1)^2 - 3(1cos(3πx) + 1) - 1) + 2)/(-1(5(1cos(3πx)... | 63 | MATH |
[
{
"content": "How many solutions does the equation (3(3(3|(-2cos(2πx - 3)) + 1| - 2)^2 - 3(3|(-2cos(2πx - 3)) + 1| - 2) - 5) - 2)/(1(3(3|(-2cos(2πx - 3)) + 1| - 2)^2 - 3(3|(-2cos(2πx - 3)) + 1| - 2) - 5)) = -1|(-3e^(-2((2x)/(1x))) + 3) + 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTe... | 38 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-1(4(5x^2 - 1x - 4) + 1) - 5)^2 - 5(-1(4(5x^2 - 1x - 4) + 1) - 5) + 4 and g(x) = (-1(2cos(2π(5x^2 + 4) + 3) - 2) + 2)/(3(2cos(2π(5x^2 + 4) + 3) - 2) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Y... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3((-1(3sin(1π((3x)/(-3x - 1)) + 1) + 1) + 2)/(-2(3sin(1π((3x)/(-3x - 1)) + 1) + 1) + 3)) + 1)/(-2((-1(3sin(1π((3x)/(-3x - 1)) + 1) + 1) + 2)/(-2(3sin(1π((3x)/(-3x - 1)) + 1) + 1) + 3)) + 3) and g(x) = -3(-1|((-1x + 2)/(2x + ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(2(1(3cos(2πx - 3) - 3) + 3) - 2) + 3 and g(x) = 2(2cos(2π((3x)/(-2x - 3)) - 2)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 49 | MATH |
[
{
"content": "How many solutions does the equation 2(2(2cos(2π(-2|x + 1|) - 3) + 2)^2 - 4(2cos(2π(-2|x + 1|) - 3) + 2) + 1) - 4 = -3(3(-1|x + 1|)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(1(-2(-5x^2 + 5x + 1) - 3) + 2) - 2)/(1(1(-2(-5x^2 + 5x + 1) - 3) + 2) - 2) and g(x) = -3(-1cos(1π((-1x - 2)/(3x + 1)) + 1) + 1)^2 - 4(-1cos(1π((-1x - 2)/(3x + 1)) + 1) + 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex form... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2e^(1((1(5(-5x + 5)) - 1)/(-1(5(-5x + 5))))) + 1 = -5(-3|(1e^(-2x) - 2) + 1| + 2)^2 - 5(-3|(1e^(-2x) - 2) + 1| + 2) have for x 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) = (-2((-3((-1(-3|x - 3| + 3) - 2)/(-2(-3|x - 3| + 3))) - 1)/(1((-1(-3|x - 3| + 3) - 2)/(-2(-3|x - 3| + 3))) - 2)) - 3)/(2((-3((-1(-3|x - 3| + 3) - 2)/(-2(-3|x - 3| + 3))) - 1)/(1((-1(-3|x - 3| + 3) - 2)/(-2(-3|x - 3| + 3))) - 2)) - 3... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1(3sin(3π((-3x + 3)/(-3x - 1)) - 2) + 2) - 5))/(-1(1(3sin(3π((-3x + 3)/(-3x - 1)) - 2) + 2) - 5) - 2) and g(x) = -3|((1(5x - 1))/(-2(5x - 1) - 2)) + 3| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed... | 35 | MATH |
[
{
"content": "How many solutions does the equation -1|(-2sin(1π(-5(-2|x - 3| + 2)) + 3) + 3)| + 2 = 3(-3cos(2π(-1x^2 + 4x + 2)) - 1)^2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 162 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-2(-2(4x^2 + 3x - 5)^2 + 2(4x^2 + 3x - 5) - 5)^2 + 2(-2(4x^2 + 3x - 5)^2 + 2(4x^2 + 3x - 5) - 5) + 2) - 3| - 3 and g(x) = -5(3(1sin(3πx + 2) - 1)^2 - 2)^2 + 2(3(1sin(3πx + 2) - 1)^2 - 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-3sin(3π(-5(5x^2 - 4x - 1)) - 3) + 1)^2 - 2(-3sin(3π(-5(5x^2 - 4x - 1)) - 3) + 1) + 1 and g(x) = -3cos(3π(1((-3x - 1)/(1x + 3)) + 5) + 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 152 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-2(-4(-2|x - 1| + 2)^2 + 1(-2|x - 1| + 2) + 3) + 1) - 1)/(2(-2(-4(-2|x - 1| + 2)^2 + 1(-2|x - 1| + 2) + 3) + 1) - 1) and g(x) = -3|((-3(-2sin(2πx - 1) - 3) + 3)/(-1(-2sin(2πx - 1) - 3) + 2)) + 1| + 2 where -10 ≤ x ≤ 10.\n... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(1(2cos(2π((1x - 1)/(1x)) - 2) + 1)^2 - 1(2cos(2π((1x - 1)/(1x)) - 2) + 1) - 2) and g(x) = 3sin(3π(-3((-2x - 2)/(1x)) - 1)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "How many solutions does the equation (-3(2|(2(2|x + 2|) - 5) + 2|) + 2)/(-1(2|(2(2|x + 2|) - 5) + 2|) + 2) = 2|((2(4x + 4) + 1)/(-1(4x + 4) - 1)) - 2| + 3 have for x 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) = (-2((-1(1sin(3π(3|x| + 2) + 1)) + 2)/(2(1sin(3π(3|x| + 2) + 1)) - 1)) + 1)/(-2((-1(1sin(3π(3|x| + 2) + 1)) + 2)/(2(1sin(3π(3|x| + 2) + 1)) - 1)) - 3) and g(x) = 3|(1|(1|x - 3| - 1) + 3| + 2) + 1| + 2 intersect in the interval [-10, 10].\n\nPresent the ... | 95 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(2π(-1|((3((3x + 3)/(3x)) + 3)/(-3((3x + 3)/(3x)) + 2)) + 2| - 2) + 1) + 2 and g(x) = (-3(-3sin(2π(-2|x - 3| + 1) - 2) - 2))/(-3(-3sin(2π(-2|x - 3| + 1) - 2) - 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format... | 193 | MATH |
[
{
"content": "How many solutions does the equation -1(3|(-3|(1x + 5) + 3| + 3)| + 1)^2 + 5(3|(-3|(1x + 5) + 3| + 3)| + 1) - 1 = -1|(3((-2x + 2)/(1x + 3)) + 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": "Determine the number of intersection points between the functions f(x) = (-1(1(-2cos(3π(-4x + 5) + 2) + 1)^2 + 2(-2cos(3π(-4x + 5) + 2) + 1) - 3) - 3)/(-1(1(-2cos(3π(-4x + 5) + 2) + 1)^2 + 2(-2cos(3π(-4x + 5) + 2) + 1) - 3) - 1) and g(x) = (1(2|((-1x + 3)/(3x - 2)) + 1| + 2) - 1)/(1(2|((-1x + 3)/(... | 230 | MATH |
[
{
"content": "How many solutions does the equation -3(-3((1((-2x)/(-1x - 2)) + 2)/(-2((-2x)/(-1x - 2)) + 3))^2 - 5((1((-2x)/(-1x - 2)) + 2)/(-2((-2x)/(-1x - 2)) + 3)) - 3)^2 - 5(-3((1((-2x)/(-1x - 2)) + 2)/(-2((-2x)/(-1x - 2)) + 3))^2 - 5((1((-2x)/(-1x - 2)) + 2)/(-2((-2x)/(-1x - 2)) + 3)) - 3) - 1 = -5((-2(1si... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((2(1|(5x^2 + 3) + 1| - 1) - 1)/(-2(1|(5x^2 + 3) + 1| - 1) - 2))^2 - 1((2(1|(5x^2 + 3) + 1| - 1) - 1)/(-2(1|(5x^2 + 3) + 1| - 1) - 2)) + 4 and g(x) = 3(5(-5x - 5)^2 - 3(-5x - 5) - 5)^2 + 4(5(-5x - 5)^2 - 3(-5x - 5) - 5) + 1 ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(1π(2(3|(-1|x + 2| - 3) - 2| - 3)^2 + 2(3|(-1|x + 2| - 3) - 2| - 3) - 3) + 1) and g(x) = (-1(1(1|x - 1| - 2) + 2) - 2)/(1(1(1|x - 1| - 2) + 2) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 1435 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(2π(-1(4(-4x + 1) - 5) + 2) - 3) - 3 and g(x) = -1((2(-1e^(-1x) + 1) - 3)/(3(-1e^(-1x) + 1) + 3)) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 607 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((-1(-2((-3x)/(1x + 1))^2 - 2((-3x)/(1x + 1)) + 5) - 3)/(1(-2((-3x)/(1x + 1))^2 - 2((-3x)/(1x + 1)) + 5) - 2)) - 3| + 3 and g(x) = 1(2cos(3π(-2|x + 1| + 3) - 2) + 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex fo... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((-2(1(2x^2 + 4x)) - 3)/(-3(1(2x^2 + 4x)) - 3))^2 - 3((-2(1(2x^2 + 4x)) - 3)/(-3(1(2x^2 + 4x)) - 3)) + 2 and g(x) = 2(-5(1x + 5) - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(1π(-1|(-3(-2|x| + 2)^2 + 2(-2|x| + 2) - 3) - 3| - 1) - 2) and g(x) = 2((-2(-2|x|) + 3)/(-2(-2|x|) + 3)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 823 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(4(-3sin(1π(-3x)) + 1) - 2) + 3)/(-2(4(-3sin(1π(-3x)) + 1) - 2) + 1) and g(x) = -4((2(-4x - 4) + 2)/(1(-4x - 4) + 3)) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 56 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-1cos(2π(1|(4x - 4) + 1|) + 3) - 3)| - 2 and g(x) = 4((2(-4x^2 + 1x + 2) + 3)/(1(-4x^2 + 1x + 2)))^2 + 4((2(-4x^2 + 1x + 2) + 3)/(1(-4x^2 + 1x + 2))) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(4(-3sin(1π(-4x^2 + 2x) + 2) - 2)^2 + 3) + 3 and g(x) = (-1(-4(-2|x + 2|)^2 - 5(-2|x + 2|) - 1) - 2)/(2(-4(-2|x + 2|)^2 - 5(-2|x + 2|) - 1) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-2cos(1π(-3(1x^2 + 3x + 4) + 5)) + 1)^2 + 4(-2cos(1π(-3(1x^2 + 3x + 4) + 5)) + 1) + 5 and g(x) = (-2(-2|((-2x + 1)/(-1x - 1)) - 2| + 1) + 3)/(1(-2|((-2x + 1)/(-1x - 1)) - 2| + 1) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the an... | 1045 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(-3|(2(2|x + 3| - 2)^2 - 1(2|x + 3| - 2) + 2)| + 1) - 2) + 3 and g(x) = 3sin(1π(4((3x + 1)/(3x + 3)) - 1) + 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 942 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1|(4(2x + 1)^2 + 5(2x + 1) - 5) + 1| - 3)^2 - 3(1|(4(2x + 1)^2 + 5(2x + 1) - 5) + 1| - 3) and g(x) = (1(2|(-2e^(1x) + 2) + 3| - 2) - 1)/(2(2|(-2e^(1x) + 2) + 3| - 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(2((3((-1x - 1)/(2x + 3)) - 3)/(1((-1x - 1)/(2x + 3))))^2 + 1((3((-1x - 1)/(2x + 3)) - 3)/(1((-1x - 1)/(2x + 3)))) - 3) + 2| + 1 and g(x) = 5(-1|((-3x - 1)/(-3x + 4)) + 2| + 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(1((3(3|x - 3| + 2) - 1)/(1(3|x - 3| + 2) - 2)) - 1) + 2)/(-3(1((3(3|x - 3| + 2) - 1)/(1(3|x - 3| + 2) - 2)) - 1) - 3) and g(x) = 1(-2|(2cos(1πx + 1)) - 1| + 1)^2 - 3(-2|(2cos(1πx + 1)) - 1| + 1) - 1 in the range -10 ≤ x ≤ 10?\n... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-1(-2|(-2sin(1πx - 2) + 1)|)^2 - 1(-2|(-2sin(1πx - 2) + 1)|) + 3) + 3| + 3 and g(x) = -5(1(-1e^(-1x) + 1)^2 - 3) - 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(2(4(-1e^(1x))^2 - 3(-1e^(1x)) - 5)^2 - 1(4(-1e^(1x))^2 - 3(-1e^(1x)) - 5) - 5)^2 + 3(2(4(-1e^(1x))^2 - 3(-1e^(1x)) - 5)^2 - 1(4(-1e^(1x))^2 - 3(-1e^(1x)) - 5) - 5) + 2 = -1(2sin(3π(-3x - 4)))^2 - 3(2sin(3π(-3x - 4))) have for x in the interval [-10, 10]?\n\n... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2e^(1(-5(-1|(-2|x + 3| + 2) + 3| + 2) - 2)) - 2 and g(x) = -1sin(1π(2((-2x + 1)/(-1x + 3))) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(1|((1(-1x^2 + 1x + 2) + 2)/(3(-1x^2 + 1x + 2) + 1)) - 3| - 1) - 1) + 1 = (1(-3cos(2π(1x^2 + 3x + 2) + 3) - 2))/(1(-3cos(2π(1x^2 + 3x + 2) + 3) - 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 29 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1e^(-2(-3(-1|(1x^2 - 5x + 5) - 1| + 2) + 3)) + 2 and g(x) = 5(-1|(-3x + 2) + 3| + 3) + 5 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|(-2cos(1π(2|(3x^2 + 2x + 5) + 1| - 3) + 1) + 2) + 3| + 1 and g(x) = (-1((3(3cos(3πx) - 2) - 3)/(-3(3cos(3πx) - 2) - 3)) + 1)/(-1((3(3cos(3πx) - 2) - 3)/(-3(3cos(3πx) - 2) - 3)) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answe... | 101 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|((-1(-3(-3e^(-2x) - 3) - 2) + 3)/(-3(-3(-3e^(-2x) - 3) - 2) + 2)) + 3| + 1 and g(x) = (2(3|(4x^2 - 4x)| - 2) - 1)/(1(3|(4x^2 - 4x)| - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2(-2sin(1π(1(1|x| + 1) + 1) - 3)) + 2)/(-3(-2sin(1π(1(1|x| + 1) + 1) - 3))) = 5(-3sin(2π(-4x^2 - 3x - 1) + 3) + 3)^2 + 1(-3sin(2π(-4x^2 - 3x - 1) + 3) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 252 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(3sin(1π(-2(3|x - 1| + 2) - 1) - 2) - 1) - 3 and g(x) = -2(-3sin(3π(-5x^2 + 2x - 1)) - 1)^2 + 2(-3sin(3π(-5x^2 + 2x - 1)) - 1) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 717 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((2(-1e^(2((2x - 3)/(1x + 1))) + 1) + 2)/(1(-1e^(2((2x - 3)/(1x + 1))) + 1) + 1)) + 2)/(-3((2(-1e^(2((2x - 3)/(1x + 1))) + 1) + 2)/(1(-1e^(2((2x - 3)/(1x + 1))) + 1) + 1)) + 1) and g(x) = (-1((3(-2sin(3πx - 3)) + 2)/(1(-2sin(3πx ... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(4(3cos(1π((3x + 2)/(3x + 3)) + 3) - 3)^2 + 5(3cos(1π((3x + 2)/(3x + 3)) + 3) - 3) - 4)^2 + 2(4(3cos(1π((3x + 2)/(3x + 3)) + 3) - 3)^2 + 5(3cos(1π((3x + 2)/(3x + 3)) + 3) - 3) - 4) - 2 and g(x) = 2cos(3π(2|(-3x - 1) + 2| - 3... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-4((2(-3x - 5) + 1)/(1(-3x - 5) + 2)))^2 + 3(-4((2(-3x - 5) + 1)/(1(-3x - 5) + 2))) - 2 and g(x) = -4(-1|(1sin(1πx)) - 3| - 3)^2 + 3(-1|(1sin(1πx)) - 3| - 3) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: ... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-1cos(2π(-2(-1x - 3)^2 + 2(-1x - 3) - 3)))^2 - 3(-1cos(2π(-2(-1x - 3)^2 + 2(-1x - 3) - 3))) - 3 and g(x) = -1|(-2|((-2x)/(-1x + 2)) + 1| + 3) - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 79 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-3sin(3π(-5(-5x^2 - 2x + 2)^2 + 4(-5x^2 - 2x + 2) + 2) - 1))| + 1 and g(x) = 1(-2(3e^(-1x))^2 + 5(3e^(-1x)) - 4) + 4 intersect 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(2|x + 2| - 1) + 1)/(1(2|x + 2| - 1) + 3)) + 4)^2 - 4(-3((-3(2|x + 2| - 1) + 1)/(1(2|x + 2| - 1) + 3)) + 4) + 1 and g(x) = 3|(-3((-3x + 3)/(-2x + 3))^2 - 1) + 1| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(3|((2(2x + 2) - 3)/(-2(2x + 2) + 3)) - 2|) - 1)/(-2(3|((2(2x + 2) - 3)/(-2(2x + 2) + 3)) - 2|) - 1) and g(x) = -3(3cos(2π(5x^2 - 3x + 3) + 3) - 2) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{You... | 1498 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-5((-3(5x - 1) - 2)/(2(5x - 1) - 1))^2 + 5((-3(5x - 1) - 2)/(2(5x - 1) - 1)) - 1) - 1| and g(x) = -5(3|(-1|x - 1| + 3)| - 3) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-3(-4((2(1x + 3) - 2)/(-1(1x + 3) + 2))^2 - 2((2(1x + 3) - 2)/(-1(1x + 3) + 2)) - 1) - 2)/(3(-4((2(1x + 3) - 2)/(-1(1x + 3) + 2))^2 - 2((2(1x + 3) - 2)/(-1(1x + 3) + 2)) - 1) - 1) = 2sin(2π(2(1x - 5)^2 - 1(1x - 5) + 4)) - 1 have for x in the interval [-10, 10]... | 990 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(3π(2|(-1(5x + 1) + 3) - 2|)) + 1 and g(x) = (2(3|(3x + 2) + 1| - 1) - 3)/(-2(3|(3x + 2) + 1| - 1) - 1) 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) = 4(2(-3cos(2π(-1x - 4) - 1) + 3) - 3)^2 + 4(2(-3cos(2π(-1x - 4) - 1) + 3) - 3) + 2 and g(x) = 2(-5(-1|x - 2| + 3) + 1)^2 + 5(-5(-1|x - 2| + 3) + 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 24 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(2π(-3|(-5(2|x - 2| - 2) - 2) - 2| - 3) - 1) and g(x) = 3((1(2|x - 2| + 2))/(1(2|x - 2| + 2) + 1))^2 - 1((1(2|x - 2| + 2))/(1(2|x - 2| + 2) + 1)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 994 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(2(4(-3|x + 3| - 1) - 4) + 4) - 1)/(-2(2(4(-3|x + 3| - 1) - 4) + 4) + 3) and g(x) = 3(3cos(2π(-2x^2 + 5x + 4) - 3) - 3)^2 + 5(3cos(2π(-2x^2 + 5x + 4) - 3) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Yo... | 771 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1((2((3(-5x^2 + 4x - 1) - 1)/(-1(-5x^2 + 4x - 1) + 2)) + 2)/(-1((3(-5x^2 + 4x - 1) - 1)/(-1(-5x^2 + 4x - 1) + 2)) - 3))^2 + 4((2((3(-5x^2 + 4x - 1) - 1)/(-1(-5x^2 + 4x - 1) + 2)) + 2)/(-1((3(-5x^2 + 4x - 1) - 1)/(-1(-5x^2 + 4x - 1... | 4 | MATH |
[
{
"content": "How many solutions does the equation (3(-2e^(2(2|(-2x + 3) - 1| + 1)) + 2))/(1(-2e^(2(2|(-2x + 3) - 1| + 1)) + 2) + 1) = -4(3cos(3π(2x^2 - 2x + 3)) + 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1066 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(5(-3(-3cos(3πx))^2 - 3(-3cos(3πx)) + 1)^2 + 5(-3(-3cos(3πx))^2 - 3(-3cos(3πx)) + 1) + 1) + 1)/(-3(5(-3(-3cos(3πx))^2 - 3(-3cos(3πx)) + 1)^2 + 5(-3(-3cos(3πx))^2 - 3(-3cos(3πx)) + 1) + 1) - 3) and g(x) = (2((3(2x + 2) + 3)... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(-4(-1(-1x^2 + 3x + 3) + 4)^2 + 4(-1(-1x^2 + 3x + 3) + 4) + 3)^2 + 1(-4(-1(-1x^2 + 3x + 3) + 4)^2 + 4(-1(-1x^2 + 3x + 3) + 4) + 3) - 4 = (-1(2(-3sin(2πx - 2) - 1) - 3))/(-1(2(-3sin(2πx - 2) - 1) - 3) + 2) have for x in the interval [-10, 10]?\n\nPresent the an... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-2|(-2cos(2π(1x^2 - 1) + 3) - 1)| + 3) + 1)/(1(-2|(-2cos(2π(1x^2 - 1) + 3) - 1)| + 3)) and g(x) = -2e^(-1((2(2x^2 + 4x + 4))/(-1(2x^2 + 4x + 4) - 2))) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed... | 54 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-3|((-2(1|x - 1| - 2) - 3)/(-2(1|x - 1| - 2) + 1)) + 1|)^2 + 4 and g(x) = 2cos(1π(1(5x + 5)^2 + 4(5x + 5)) - 3) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(1sin(2π((-1(2|x - 2| + 1) + 2)/(1(2|x - 2| + 1) + 1)) + 1) + 1)^2 + 3(1sin(2π((-1(2|x - 2| + 1) + 2)/(1(2|x - 2| + 1) + 1)) + 1) + 1) - 1 and g(x) = -3((-1(1|x - 3|) - 1)/(-2(1|x - 3|) + 1)) + 2 in the range -10 ≤ x ≤ 10?\n\nPre... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(1(-3|(-1x^2 + 3x + 2) + 1|)^2 - 3(-3|(-1x^2 + 3x + 2) + 1|) - 2) - 3| + 3 and g(x) = -4(-1cos(3π(1|x - 2| - 1)) + 3)^2 - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 4 | MATH |
[
{
"content": "How many solutions does the equation 4((3(2e^(-2((2x + 2)/(-1x + 3))) + 3) - 1)/(-3(2e^(-2((2x + 2)/(-1x + 3))) + 3) - 1)) - 5 = (-1(-1|(2x^2 + 1x) - 3| + 3) + 1)/(-3(-1|(2x^2 + 1x) - 3| + 3) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-2|(1|(-2|x - 1| + 3) + 3| + 2) + 1| + 3) - 4 and g(x) = 3|((1(-3|x + 2|))/(-3(-3|x + 2|) + 4)) - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation -1((1(1(-1x - 1)^2 + 1(-1x - 1) - 4) + 2)/(1(1(-1x - 1)^2 + 1(-1x - 1) - 4)))^2 - 4((1(1(-1x - 1)^2 + 1(-1x - 1) - 4) + 2)/(1(1(-1x - 1)^2 + 1(-1x - 1) - 4))) + 2 = 1sin(3π(1(1x - 3)^2 + 1(1x - 3) + 1) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent t... | 30 | MATH |
[
{
"content": "How many solutions does the equation (1((1(2|(-2|x| + 3)| - 1) + 1)/(-1(2|(-2|x| + 3)| - 1) + 2)) - 3)/(-2((1(2|(-2|x| + 3)| - 1) + 1)/(-1(2|(-2|x| + 3)| - 1) + 2)) + 3) = 1(-3cos(2π(-2x + 3) - 2) - 2)^2 + 1(-3cos(2π(-2x + 3) - 2) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answe... | 152 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(3e^(1(2(-3|x + 3| - 2) + 5)) - 2) - 2)/(3(3e^(1(2(-3|x + 3| - 2) + 5)) - 2)) and g(x) = -1sin(1π(4(-2|x - 3|)^2 - 5) - 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 732 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(3π(-3(-1|(-2x^2 - 2) - 3| - 2) + 1) - 1) - 1 and g(x) = 5(-1sin(2π((1x)/(-2x + 3)) - 3) - 2)^2 + 5(-1sin(2π((1x)/(-2x + 3)) - 3) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 295 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(3sin(2π(2|((-2x + 2)/(-1x + 3)) - 3| - 3) + 1) + 2) + 5 and g(x) = -5(1(-3e^(1x) + 3) - 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) = (2(-3|((2(-2sin(2πx - 3) - 1) - 3)/(-1(-2sin(2πx - 3) - 1) - 1)) - 3| + 1) + 1)/(3(-3|((2(-2sin(2πx - 3) - 1) - 3)/(-1(-2sin(2πx - 3) - 1) - 1)) - 3| + 1)) and g(x) = -1|(-4((3x + 3)/(-3x + 2)) + 5) - 3| + 1 in the range -10 ≤ x ≤ ... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π(5(1|(-1|x - 3| - 3) + 1| - 3) - 4) + 2) - 1 and g(x) = -2|((-1((2x - 2)/(1x + 2)) + 3)/(1((2x - 2)/(1x + 2)) - 3)) + 2| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 51 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-1((-2(-3e^(2x) + 1) - 1)/(3(-3e^(2x) + 1) + 1))^2 + 4((-2(-3e^(2x) + 1) - 1)/(3(-3e^(2x) + 1) + 1)) - 4) and g(x) = (1(-5(-1cos(2πx - 3)) - 2) - 1)/(1(-5(-1cos(2πx - 3)) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex forma... | 16 | MATH |
[
{
"content": "How many solutions does the equation 4(-4((1(-3x + 3) + 2)/(-2(-3x + 3) - 1)) - 5)^2 + 1(-4((1(-3x + 3) + 2)/(-2(-3x + 3) - 1)) - 5) + 1 = 1|(2e^(1(4x^2 - 2x - 3)) + 1) + 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2cos(3π(2|(-2(3|x - 2| + 3) - 1) - 2|) + 3) + 2 = 3sin(3π(-2|(-3x + 2) + 3| + 3) - 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 360 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1((3(5(-1x - 1) - 1) + 1)/(1(5(-1x - 1) - 1))) - 2)/(-3((3(5(-1x - 1) - 1) + 1)/(1(5(-1x - 1) - 1)))) and g(x) = 4(3sin(3π(-1|x - 2| + 2) + 2) - 1) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{You... | 61 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5((-1(-2sin(3π(-4x - 5) + 2) - 3) + 1)/(3(-2sin(3π(-4x - 5) + 2) - 3) + 3))^2 + 5 and g(x) = 1(-5(1sin(1πx - 3) + 3)^2 - 4(1sin(1πx - 3) + 3) - 1)^2 - 5(-5(1sin(1πx - 3) + 3)^2 - 4(1sin(1πx - 3) + 3) - 1) - 3 intersect in the interval [-10, 10].\n\nPre... | 173 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2sin(2π(-3|((-2x - 3)/(2x + 1))| - 3) + 1) + 2)^2 - 4(2sin(2π(-3|((-2x - 3)/(2x + 1))| - 3) + 1) + 2) + 2 and g(x) = 2sin(1π(-5(-4x + 2) + 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 237 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-3cos(3π(-3|(-4x - 3)| - 3) + 2) + 2)^2 + 5 and g(x) = -2|(2|(-1cos(2πx - 2) - 1) - 3| - 1) + 1| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 719 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(3π(5(4((-2x - 1)/(3x + 3))^2 + 5((-2x - 1)/(3x + 3)) + 3)^2 - 1(4((-2x - 1)/(3x + 3))^2 + 5((-2x - 1)/(3x + 3)) + 3) + 3)) - 3 and g(x) = (-2(-2(-5x - 1)^2 - 1(-5x - 1) - 3) + 1)/(1(-2(-5x - 1)^2 - 1(-5x - 1) - 3) + 3) ... | 152 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-4(2|(1cos(3πx + 2) - 3) - 2| - 1)^2 - 3(2|(1cos(3πx + 2) - 3) - 2| - 1) - 1) + 3)/(-2(-4(2|(1cos(3πx + 2) - 3) - 2| - 1)^2 - 3(2|(1cos(3πx + 2) - 3) - 2| - 1) - 1) + 3) and g(x) = 1(3(-1|x + 3| + 3) + 2)^2 - 5(3(-1|x + 3... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(3cos(3π(1|(-2|x + 1| + 3) - 3| + 2) + 3) - 1) + 3 and g(x) = 1(-2e^(1((-1x - 3)/(-1x))) + 1) + 5 intersect 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(1|(-5((-1x)/(-2x - 2))^2 + 3((-1x)/(-2x - 2)) + 5) + 3| - 2) - 1)/(3(1|(-5((-1x)/(-2x - 2))^2 + 3((-1x)/(-2x - 2)) + 5) + 3| - 2) - 1) and g(x) = (-1((2(-2x^2 - 3x - 4) - 3)/(3(-2x^2 - 3x - 4) + 1)) - 1)/(1((2(-2x^2 - 3x ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(4(-3cos(3π(-5x^2 + 2x + 4) + 1) - 3) + 4) + 5 and g(x) = -4(1|(-2|x| - 3) - 1| - 1) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1280 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(2(2(-5x) - 3)^2 + 1) + 3)/(3(2(2(-5x) - 3)^2 + 1) - 2) and g(x) = 2sin(1π(-2(-1x + 3) + 4) - 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many solutions does the equation -1(2(1(5x^2 - 1x + 1)^2 - 5(5x^2 - 1x + 1) - 4) - 5)^2 + 4(2(1(5x^2 - 1x + 1)^2 - 5(5x^2 - 1x + 1) - 4) - 5) + 1 = (2((-2(1x - 3))/(-1(1x - 3) - 3)) - 2)/(-1((-2(1x - 3))/(-1(1x - 3) - 3))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex fo... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(1(-1|(2x + 3) + 1| - 3)^2 + 3(-1|(2x + 3) + 1| - 3) - 2) + 3| + 3 and g(x) = -3(-3sin(3π(-4x^2 - 2x + 5) - 3) - 3)^2 - 4(-3sin(3π(-4x^2 - 2x + 5) - 3) - 3) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \... | 209 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-1(5(3e^(1x))^2 + 2(3e^(1x)) - 2)^2 + 5(5(3e^(1x))^2 + 2(3e^(1x)) - 2) + 5) - 3 and g(x) = 3|(2sin(1π(-2|x + 2| + 2) + 1) + 3) - 2| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(1π(3|((1(-4x - 2) - 3)/(-1(-4x - 2) - 1)) + 3| + 2) - 2) and g(x) = -3((-3(2sin(2πx - 3) + 1) + 3)/(-3(2sin(2πx - 3) + 1) - 2))^2 - 3((-3(2sin(2πx - 3) + 1) + 3)/(-3(2sin(2πx - 3) + 1) - 2)) + 1 in the range -10 ≤ x ≤ 10?\n\nP... | 93 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-1|(-3(3x^2 - 1x - 2)^2 + 1(3x^2 - 1x - 2) + 2) - 1| - 1) - 2| - 2 and g(x) = -1sin(1π((2(-2|x + 2| + 1) + 3)/(-1(-2|x + 2| + 1) + 3))) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
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