messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(2π(-2|(4(-4x - 1) + 3) - 1| + 2)) and g(x) = (-3(4(1x + 3) + 2) - 1)/(-2(4(1x + 3) + 2) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1008 | MATH |
[
{
"content": "How many solutions does the equation 5(1sin(2π(1((3x + 1)/(2x + 2))^2 + 1((3x + 1)/(2x + 2)) + 1)) - 1)^2 + 4(1sin(2π(1((3x + 1)/(2x + 2))^2 + 1((3x + 1)/(2x + 2)) + 1)) - 1) - 4 = 1|(-1((-1x - 2)/(3x))) - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your... | 87 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2|(-1e^(-2((-2x)/(-3x + 4))) + 2) - 2| - 2)^2 + 4(2|(-1e^(-2((-2x)/(-3x + 4))) + 2) - 2| - 2) + 2 and g(x) = 2cos(3π(5(-5x^2 - 4x - 4)^2 + 1(-5x^2 - 4x - 4) + 1) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\... | 46 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(1|(2(-1x^2 - 3x - 2) - 3) + 3|) - 3| - 2 and g(x) = 2sin(3π(3|(-2x - 1)|) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π(-2(-1|(-1x - 4) + 3| + 1)^2 + 2(-1|(-1x - 4) + 3| + 1) + 2) - 3) - 3 and g(x) = 4((1(-4x - 2) + 3)/(-3(-4x - 2)))^2 + 3((1(-4x - 2) + 3)/(-3(-4x - 2))) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\... | 364 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(1(-1(1x^2 + 5x)^2 + 5) + 1) - 5 and g(x) = -3((2(-3|x| + 3))/(2(-3|x| + 3) + 3)) - 5 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) = 4(-3(4(2|x| - 1) - 2) + 3) + 3 and g(x) = -4(3sin(2π((-1x + 3)/(2x + 1)) - 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many solutions does the equation -1(-3|(-1|(3|x + 2| - 2) + 3| - 1) + 3| - 3)^2 - 4(-3|(-1|(3|x + 2| - 2) + 3| - 1) + 3| - 3) + 4 = 1(3((-2x - 1)/(-3x + 1))^2 - 2((-2x - 1)/(-3x + 1)) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(2|(-2(-3cos(2πx - 2) + 3) - 2) - 2| - 3) - 3| - 2 and g(x) = 1|(-3(-1|x - 3| + 2)^2 - 5(-1|x - 3| + 2) + 3)| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2e^(2(-3(3|(-2x^2 - 3x - 1) - 3| - 2)^2 + 4(3|(-2x^2 - 3x - 1) - 3| - 2) + 3)) - 1 and g(x) = -3(-1(-4x^2 + 3x - 1)^2 + 1(-4x^2 + 3x - 1) + 5)^2 + 1(-1(-4x^2 + 3x - 1)^2 + 1(-4x^2 + 3x - 1) + 5) + 2 in the range -10 ≤ x ≤ 10?\n\nP... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(2|(2|((1x - 2)/(-3x + 2)) + 1| + 2) - 2| - 2) - 1)/(1(2|(2|((1x - 2)/(-3x + 2)) + 1| + 2) - 2| - 2) + 2) and g(x) = -3cos(3π((1(1x^2 + 1x - 5) - 2)/(-3(1x^2 + 1x - 5) - 1)) - 3) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answe... | 43 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-4((-3(-1|x - 2| + 3) - 1)/(1(-1|x - 2| + 3) + 3))^2 + 5((-3(-1|x - 2| + 3) - 1)/(1(-1|x - 2| + 3) + 3)) - 4) - 2| - 2 and g(x) = (2(-2sin(2π(-4x^2 - 1x + 3) + 2) + 1) - 2)/(2(-2sin(2π(-4x^2 - 1x + 3) + 2) + 1) + 1) in the rang... | 34 | MATH |
[
{
"content": "How many solutions does the equation -1|(-1(-2(-4x^2 + 5)^2 + 3(-4x^2 + 5) - 4)^2 - 1(-2(-4x^2 + 5)^2 + 3(-4x^2 + 5) - 4) + 2) + 1| - 2 = -3|((1(-2e^(-1x)) + 3)/(-2(-2e^(-1x)) - 1)) - 1| have for x 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 3(2(-1((-1x + 2)/(-2x))^2 - 4((-1x + 2)/(-2x)) + 1)^2 - 2(-1((-1x + 2)/(-2x))^2 - 4((-1x + 2)/(-2x)) + 1) - 4) + 3 = -1cos(2π((-1(-5x^2 - 3x + 5) - 2)/(2(-5x^2 - 3x + 5) - 1))) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\b... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-2cos(2π(-2((-1x + 2)/(-2x - 1)) - 4) - 3) - 2) - 2| - 1 and g(x) = 1(2sin(3π(1|x + 3|) + 2) - 2)^2 + 3(2sin(3π(1|x + 3|) + 2) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 21 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(2π(5(5((-1x + 1)/(-3x + 4))^2 - 2((-1x + 1)/(-3x + 4)) - 5))) - 1 and g(x) = -3(-2|(1x^2 - 2x + 4) - 2| + 1)^2 - 2(-2|(1x^2 - 2x + 4) - 2| + 1) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{You... | 21 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4((1(2|(-2x^2 + 2x - 4) - 2| + 1) - 2)/(-2(2|(-2x^2 + 2x - 4) - 2| + 1) + 3))^2 + 5((1(2|(-2x^2 + 2x - 4) - 2| + 1) - 2)/(-2(2|(-2x^2 + 2x - 4) - 2| + 1) + 3)) + 5 and g(x) = 1sin(1π(1((-2x - 1)/(-2x - 2)) - 5) - 3) + 3 in the rang... | 24 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|((-3(-1cos(2π(-1|x + 1|) - 1) - 3) + 2)/(-3(-1cos(2π(-1|x + 1|) - 1) - 3) + 2)) - 3| + 3 and g(x) = (-2((-1(2cos(3πx + 1) + 1) - 2)/(2(2cos(3πx + 1) + 1) - 1)) + 1)/(-2((-1(2cos(3πx + 1) + 1) - 2)/(2(2cos(3πx + 1) + 1) - 1)) - 3) intersect in the int... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(2sin(2π(1|(-4x^2 - 4x + 3) + 1| - 1) + 3) - 2) - 2| - 2 and g(x) = 2sin(3π(1(3x^2 - 1x + 3) + 2) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 757 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((-3(2|(-2|x - 1| - 1) + 2| - 1) + 1)/(-3(2|(-2|x - 1| - 1) + 2| - 1) + 4))^2 + 1((-3(2|(-2|x - 1| - 1) + 2| - 1) + 1)/(-3(2|(-2|x - 1| - 1) + 2| - 1) + 4)) + 2 and g(x) = -3((-1(1sin(2πx) + 1) - 2)/(-2(1sin(2πx) + 1) - 3))^2 + 4... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-2(1(-2cos(3πx + 2) + 3)^2 - 3(-2cos(3πx + 2) + 3) + 2) + 1) - 2)/(-2(-2(1(-2cos(3πx + 2) + 3)^2 - 3(-2cos(3πx + 2) + 3) + 2) + 1)) and g(x) = -2cos(2π((-1((-2x + 1)/(-1x + 2)) - 3)/(-1((-2x + 1)/(-1x + 2)) - 2)) + 2) + 1 intersect in the interval ... | 81 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1sin(2π(-1(2(-3x - 3)^2 - 3(-3x - 3) + 3) + 2) + 3) + 2 and g(x) = -3cos(2π(-3((-3x + 2)/(-3x + 1))^2 + 3((-3x + 2)/(-3x + 1)) - 3) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 643 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-1(-1((-3x - 3)/(-1x + 2)) - 2) - 3) - 3)/(-3(-1(-1((-3x - 3)/(-1x + 2)) - 2) - 3) - 2) and g(x) = -3cos(3π(-2(5x - 5) - 1) + 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 267 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-4(2sin(3π(-1|x|) + 3)) - 4)^2 - 4(-4(2sin(3π(-1|x|) + 3)) - 4) + 1 and g(x) = (2((-1(-1x) + 1)/(-1(-1x) + 2)))/(-3((-1(-1x) + 1)/(-1(-1x) + 2)) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 119 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-2cos(1π(-2((3x + 2)/(2x + 1))^2 + 5((3x + 2)/(2x + 1)) - 1) - 3)) + 3)/(-1(-2cos(1π(-2((3x + 2)/(2x + 1))^2 + 5((3x + 2)/(2x + 1)) - 1) - 3)) - 2) and g(x) = -5(2|(-1cos(3πx) - 1) - 1|) + 3 intersect in the interval [-10, 10].\n\nPresent the answe... | 21 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((-1(-2cos(3π(3x + 4))) + 3)/(2(-2cos(3π(3x + 4))) + 2)))/(2((-1(-2cos(3π(3x + 4))) + 3)/(2(-2cos(3π(3x + 4))) + 2)) - 1) and g(x) = -3sin(2π(4(5x - 5) + 1) + 2) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex forma... | 566 | MATH |
[
{
"content": "How many solutions does the equation 3|(-4(2(1x^2 - 2x - 2)^2 + 3(1x^2 - 2x - 2) + 1) - 2)| - 3 = (3(-5(5x + 1)) - 2)/(1(-5(5x + 1))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(3|((-3(-5x + 1) - 3)/(-3(-5x + 1) - 2)) + 1| + 3) + 1)/(3(3|((-3(-5x + 1) - 3)/(-3(-5x + 1) - 2)) + 1| + 3) + 2) and g(x) = 4(1|(4x^2 + 1x - 2)| - 2)^2 + 5(1|(4x^2 + 1x - 2)| - 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the an... | 8 | MATH |
[
{
"content": "How many solutions does the equation 1|(1|(4((-3x - 1)/(-1x)) + 4) + 3| - 2) - 2| - 1 = (-1(1sin(1π(3x^2 - 5x + 3) - 2) + 3) - 3)/(-2(1sin(1π(3x^2 - 5x + 3) - 2) + 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": "Find the number of times the graphs of f(x) = -3|((-2(-2sin(3π(-1x - 4) - 1) - 1) + 3)/(-1(-2sin(3π(-1x - 4) - 1) - 1) - 3)) + 1| + 2 and g(x) = -4(3e^(-1(2x + 4)) + 1)^2 + 1(3e^(-1(2x + 4)) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 41 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(-2(2sin(2π(4x + 5) + 3) - 1) - 2) - 3| and g(x) = -3(-4(1x - 4) + 2)^2 + 4(-4(1x - 4) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "How many solutions does the equation -1((3(2(-2|x| - 1) + 2) + 2)/(-3(2(-2|x| - 1) + 2) + 4))^2 - 5((3(2(-2|x| - 1) + 2) + 2)/(-3(2(-2|x| - 1) + 2) + 4)) - 2 = -2sin(1π(1((-1x + 3)/(2x + 2)) - 2) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 43 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2((1(-3(-2sin(2πx - 2))^2 + 4(-2sin(2πx - 2)) - 1) + 1)/(2(-3(-2sin(2πx - 2))^2 + 4(-2sin(2πx - 2)) - 1) - 3)) - 3)/(2((1(-3(-2sin(2πx - 2))^2 + 4(-2sin(2πx - 2)) - 1) + 1)/(2(-3(-2sin(2πx - 2))^2 + 4(-2sin(2πx - 2)) - 1) - ... | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(1e^(1(1(-3|x + 1| + 3)^2 + 2(-3|x + 1| + 3))) - 1) - 5 and g(x) = (2(2(3x^2 + 5x + 4)^2 - 4(3x^2 + 5x + 4) - 1) + 1)/(2(2(3x^2 + 5x + 4)^2 - 4(3x^2 + 5x + 4) - 1) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex for... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4((-3((2(-5x^2 + 4x + 3) - 1)/(3(-5x^2 + 4x + 3) + 1)))/(3((2(-5x^2 + 4x + 3) - 1)/(3(-5x^2 + 4x + 3) + 1))))^2 - 1 and g(x) = (1(-2sin(3π(2x + 5) - 1) + 1) + 2)/(-2(-2sin(3π(2x + 5) - 1) + 1) - 1) where -10 ≤ x ≤ 10.\n\nPres... | 120 | MATH |
[
{
"content": "How many solutions does the equation (-1(1|(4((-2x - 2)/(-1x + 3))^2 - 4((-2x - 2)/(-1x + 3)) - 5) + 3|) + 2)/(3(1|(4((-2x - 2)/(-1x + 3))^2 - 4((-2x - 2)/(-1x + 3)) - 5) + 3|) - 2) = 3cos(3π(3(-1|x - 3| + 3) + 5) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format... | 144 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|((2(-3(-2x + 3) + 3))/(-2(-3(-2x + 3) + 3) - 2)) - 2| + 3 and g(x) = -4(4(2cos(1πx + 3))^2 - 2(2cos(1πx + 3)) - 4) - 5 intersect 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) = -3e^(1(1(3(-4x^2 + 2x - 5) + 2) - 5)) and g(x) = 1(5(3sin(3πx + 2))^2 + 4(3sin(3πx + 2)) - 4) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "How many solutions does the equation (2(-3|((1((1x - 1)/(-1x + 2)) + 3)/(-2((1x - 1)/(-1x + 2)) + 3))|))/(3(-3|((1((1x - 1)/(-1x + 2)) + 3)/(-2((1x - 1)/(-1x + 2)) + 3))|) - 2) = -1(-5(-3sin(1πx) - 2)^2 + 4(-3sin(1πx) - 2) + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaT... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-2(4((-3x - 2)/(-2x - 3)) - 5)^2 + 1(4((-3x - 2)/(-2x - 3)) - 5) + 3)^2 - 5(-2(4((-3x - 2)/(-2x - 3)) - 5)^2 + 1(4((-3x - 2)/(-2x - 3)) - 5) + 3) - 5 and g(x) = -1sin(2π(-2|(-3x + 2) + 1| - 3) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent ... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3((2(-1(-5x + 4)^2 - 1) - 1)/(3(-1(-5x + 4)^2 - 1) + 2)) + 3)/(1((2(-1(-5x + 4)^2 - 1) - 1)/(3(-1(-5x + 4)^2 - 1) + 2)) - 3) and g(x) = -5(2|(3cos(2πx + 3) - 2) + 3|) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex f... | 80 | MATH |
[
{
"content": "How many solutions does the equation -1(-1((-2(-3e^(2x) + 2) - 3)/(2(-3e^(2x) + 2) - 1))^2 - 5((-2(-3e^(2x) + 2) - 3)/(2(-3e^(2x) + 2) - 1)) + 5)^2 + 4(-1((-2(-3e^(2x) + 2) - 3)/(2(-3e^(2x) + 2) - 1))^2 - 5((-2(-3e^(2x) + 2) - 3)/(2(-3e^(2x) + 2) - 1)) + 5) - 2 = -2(3(1|x| - 3) - 4)^2 - 2(3(1|x| -... | 6 | MATH |
[
{
"content": "How many solutions does the equation 1(1(4((-1x - 2)/(-3x - 1))) + 5) - 1 = (3(-3sin(1π(-1|x + 2| + 1) + 1) - 2) - 1)/(1(-3sin(1π(-1|x + 2| + 1) + 1) - 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many solutions does the equation 2sin(3π((-1(2(5x + 5) + 1))/(-2(2(5x + 5) + 1) + 1)) - 1) + 3 = 1|(3(-1sin(1πx + 1) + 1) + 4) - 3| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(1e^(-2(-3((-2x + 1)/(3x + 1)) - 5)) + 1) - 3| - 2 and g(x) = 3cos(2π(2|(-2|x + 1| + 1) - 1| - 2) - 1) - 3 in the range -10 ≤ x ≤ 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) = -2cos(1π(5(5((-2x + 3)/(1x + 2)) - 5)^2 + 5(5((-2x + 3)/(1x + 2)) - 5) - 5) + 1) and g(x) = -2e^(-2((2((-1x)/(3x - 2)))/(-1((-1x)/(3x - 2)) - 2))) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1420 | MATH |
[
{
"content": "How many solutions does the equation (2((-1(-4(-2x^2 + 5x + 1)^2 - 3(-2x^2 + 5x + 1) + 2) + 2)/(-3(-4(-2x^2 + 5x + 1)^2 - 3(-2x^2 + 5x + 1) + 2) - 1)))/(2((-1(-4(-2x^2 + 5x + 1)^2 - 3(-2x^2 + 5x + 1) + 2) + 2)/(-3(-4(-2x^2 + 5x + 1)^2 - 3(-2x^2 + 5x + 1) + 2) - 1)) - 1) = 1(-1((2x - 2)/(-3x - 1))^... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-1(-4(-5x + 1)) + 5) + 1)/(-2(-1(-4(-5x + 1)) + 5) - 1) and g(x) = (-1(3|(3sin(3πx - 2) - 1) + 2| - 1))/(-2(3|(3sin(3πx - 2) - 1) + 2| - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 98 | MATH |
[
{
"content": "How many solutions does the equation 2(-1((2(2|x + 2| - 2) - 3)/(1(2|x + 2| - 2)))^2 - 5((2(2|x + 2| - 2) - 3)/(1(2|x + 2| - 2))) + 4)^2 + 4(-1((2(2|x + 2| - 2) - 3)/(1(2|x + 2| - 2)))^2 - 5((2(2|x + 2| - 2) - 3)/(1(2|x + 2| - 2))) + 4) - 3 = 1(-2cos(2π(-1|x - 2| - 3) - 1) + 3) + 4 have for x in t... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2|(2sin(3π(3|x| + 1) + 3) - 1) + 3| + 1)^2 + 5(2|(2sin(3π(3|x| + 1) + 3) - 1) + 3| + 1) and g(x) = 1|((-1(3sin(3πx - 1) + 2) + 1)/(1(3sin(3πx - 1) + 2) - 2)) + 2| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex form... | 26 | MATH |
[
{
"content": "How many solutions does the equation 4(1(1|(-1x - 1) + 1| - 3)^2 + 4(1|(-1x - 1) + 1| - 3) - 4)^2 + 2(1(1|(-1x - 1) + 1| - 3)^2 + 4(1|(-1x - 1) + 1| - 3) - 4) - 4 = 3sin(1π(5(-2|x + 1| + 1)^2 - 5(-2|x + 1| + 1) - 3) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex form... | 37 | MATH |
[
{
"content": "How many solutions does the equation (2((1(-1((-3x + 1)/(3x - 2)) - 2) - 3)/(-3(-1((-3x + 1)/(3x - 2)) - 2) + 1)) + 2)/(-2((1(-1((-3x + 1)/(3x - 2)) - 2) - 3)/(-3(-1((-3x + 1)/(3x - 2)) - 2) + 1))) = (-1(4(-1cos(3πx) + 2) - 5) + 2)/(-3(4(-1cos(3πx) + 2) - 5) + 2) have for x in the interval [-10, 1... | 58 | MATH |
[
{
"content": "How many solutions does the equation -2((-2(1(1sin(1πx + 1) + 1)^2 - 4(1sin(1πx + 1) + 1) + 5) - 2)/(-1(1(1sin(1πx + 1) + 1)^2 - 4(1sin(1πx + 1) + 1) + 5) - 3))^2 - 1((-2(1(1sin(1πx + 1) + 1)^2 - 4(1sin(1πx + 1) + 1) + 5) - 2)/(-1(1(1sin(1πx + 1) + 1)^2 - 4(1sin(1πx + 1) + 1) + 5) - 3)) + 1 = 2(-1... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(5(-3cos(3π(-1x^2 + 3x + 5)) - 2) - 1)^2 - 5(5(-3cos(3π(-1x^2 + 3x + 5)) - 2) - 1) + 1 and g(x) = -5(1sin(1π(-1x - 1) - 2) - 2)^2 - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 819 | MATH |
[
{
"content": "How many solutions does the equation 1(3|(-3cos(3π((-3x)/(-2x - 2)) - 1) - 2) + 2| + 3) + 3 = 5(1(1x^2 + 4x - 1)^2 + 5(1x^2 + 4x - 1) - 4)^2 + 5(1(1x^2 + 4x - 1)^2 + 5(1x^2 + 4x - 1) - 4) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3((-1((2(-3x + 4))/(2(-3x + 4) - 3)) - 3)/(2((2(-3x + 4))/(2(-3x + 4) - 3)) + 1)) + 2)/(-1((-1((2(-3x + 4))/(2(-3x + 4) - 3)) - 3)/(2((2(-3x + 4))/(2(-3x + 4) - 3)) + 1)) + 3) and g(x) = -4(-2cos(2π(-2|x| + 2) - 2) - 1) - 5 intersect in the interval [... | 81 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((-2((-1(1cos(2πx) + 3) + 3)/(1(1cos(2πx) + 3))))/(-2((-1(1cos(2πx) + 3) + 3)/(1(1cos(2πx) + 3))) + 3)) and g(x) = -1cos(2π(-4((-2x - 1)/(1x + 3))^2 + 5((-2x - 1)/(1x + 3)) - 2)) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answ... | 186 | MATH |
[
{
"content": "How many solutions does the equation -5(3(-1e^(2((-2x - 1)/(1x - 3))) + 1) - 4)^2 + 2(3(-1e^(2((-2x - 1)/(1x - 3))) + 1) - 4) + 4 = -3(-2sin(3π(5x - 2) + 2)) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 209 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-2sin(3π(-1|(3x - 1) - 3| + 1) - 1) + 1) - 1| + 2 and g(x) = 2cos(1π(-3(1x - 3) + 4) - 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 240 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-1|((-1((-2x + 2)/(2x - 1)) + 3)/(-1((-2x + 2)/(2x - 1)) - 3))| + 3) and g(x) = (2(-2(3sin(2πx - 3) - 3)^2 - 3(3sin(2πx - 3) - 3)))/(2(-2(3sin(2πx - 3) - 3)^2 - 3(3sin(2πx - 3) - 3)) + 1) where -10 ≤ x ≤ 10.\n\nPresent the ... | 29 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-1|((-3(-2e^(1x) - 3) - 3)/(2(-2e^(1x) - 3) - 1)) - 1| - 1) - 3)/(-1(-1|((-3(-2e^(1x) - 3) - 3)/(2(-2e^(1x) - 3) - 1)) - 1| - 1) + 2) and g(x) = 4(4(3cos(1πx + 3) - 2) - 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in... | 20 | MATH |
[
{
"content": "How many solutions does the equation 3(3sin(1π(-3|(-3x - 4) - 1| + 1) + 3) - 2)^2 - 4 = -1|(-1(2|x + 2| - 2)^2 - 3(2|x + 2| - 2) + 1) + 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-1cos(2π(-4(5x^2 - 3x)^2 - 3(5x^2 - 3x) - 5)) + 2) + 3)/(-1(-1cos(2π(-4(5x^2 - 3x)^2 - 3(5x^2 - 3x) - 5)) + 2) + 2) and g(x) = -5((2(-1x^2 + 2x - 5) + 3)/(-2(-1x^2 + 2x - 5))) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer... | 74 | MATH |
[
{
"content": "How many solutions does the equation -3|(-5(-2((2x + 3)/(-2x + 3)) - 3)^2 - 3(-2((2x + 3)/(-2x + 3)) - 3) + 5) - 3| + 3 = 2|(-1(-3sin(3πx))^2 + 5(-3sin(3πx)) + 4) + 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) = -3(5(-3(-1x^2 - 4x + 5) + 5)^2 + 4(-3(-1x^2 - 4x + 5) + 5) - 2) + 3 and g(x) = -3((1((-3x)/(-1x)))/(2((-3x)/(-1x)) - 1))^2 - 1((1((-3x)/(-1x)))/(2((-3x)/(-1x)) - 1)) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex for... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-2|(-1|((-3x + 3)/(-1x - 2)) - 3| + 3)| + 3) + 2)/(2(-2|(-1|((-3x + 3)/(-1x - 2)) - 3| + 3)| + 3) + 1) and g(x) = (2(-3sin(3π((-2x - 1)/(-2x)) + 2) + 1) - 2)/(-2(-3sin(3π((-2x - 1)/(-2x)) + 2) + 1)) intersect in the interval [-10, 10].\n\nPresent t... | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1(-1(2|x| + 3)^2 + 2(2|x| + 3) - 4) + 4) + 2)/(3(1(-1(2|x| + 3)^2 + 2(2|x| + 3) - 4) + 4) + 3) and g(x) = (-2(1cos(3π(2|x + 3| + 2)) + 2) + 3)/(-1(1cos(3π(2|x + 3| + 2)) + 2) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer ... | 120 | MATH |
[
{
"content": "How many solutions does the equation 5(4((-1(3|x + 2| - 2) - 3)/(-3(3|x + 2| - 2) - 2))^2 - 1((-1(3|x + 2| - 2) - 3)/(-3(3|x + 2| - 2) - 2)) + 3) + 3 = 4(-2|(1x^2 + 4x - 4) + 3| - 3)^2 + 5(-2|(1x^2 + 4x - 4) + 3| - 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format:... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2(2cos(3π(1|((-1x - 2)/(-3x + 4))| + 3) + 1) + 1))/(-2(2cos(3π(1|((-1x - 2)/(-3x + 4))| + 3) + 1) + 1) + 3) = -1|(1(2|x + 1| + 2)^2 + 1(2|x + 1| + 2)) + 2| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(2sin(1π(2|(4x^2 + 2) - 1| + 3) + 2) + 1) and g(x) = -3cos(2π((3(3|x| - 2))/(-3(3|x| - 2) - 1)) + 3) + 3 in the range -10 ≤ x ≤ 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) = -4(-3sin(2π(-3(3x^2 + 2x - 2) + 3)) - 3) - 2 and g(x) = 1cos(2π(-4(3|x - 2| + 3) - 2) + 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 287 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-2e^(1((-2(-3|x - 1| - 3) - 1)/(3(-3|x - 1| - 3) - 1))) - 2) + 3| and g(x) = 5((-2(3x^2 - 4x - 2) + 2)/(-2(3x^2 - 4x - 2) - 1))^2 + 2((-2(3x^2 - 4x - 2) + 2)/(-2(3x^2 - 4x - 2) - 1)) - 5 where -10 ≤ x ≤ 10.\n\nPresent the ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-2|(-3sin(1π(-3x^2 + 3x) + 2) - 3) - 1| + 2) and g(x) = (-3(-5(-4x^2 + 5x - 2) + 1) + 1)/(-3(-5(-4x^2 + 5x - 2) + 1) + 4) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 479 | MATH |
[
{
"content": "How many solutions does the equation 1sin(2π(1(2(-3|x - 2| + 3)^2 + 4(-3|x - 2| + 3) + 5)^2 - 1(2(-3|x - 2| + 3)^2 + 4(-3|x - 2| + 3) + 5) + 5)) + 2 = -2((-2(-1e^(1x)))/(1(-1e^(1x)))) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 765 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((3(-2(-2|x + 2| + 2)^2 + 4(-2|x + 2| + 2) - 2) + 2)/(2(-2(-2|x + 2| + 2)^2 + 4(-2|x + 2| + 2) - 2) + 3)) + 1| and g(x) = 2(-4((-2x)/(-2x))^2 + 3((-2x)/(-2x)) + 5)^2 - 1(-4((-2x)/(-2x))^2 + 3((-2x)/(-2x)) + 5) + 4 where -10... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-2|(4((-2x - 2)/(-2x + 1)) + 1) - 1| + 1))/(2(-2|(4((-2x - 2)/(-2x + 1)) + 1) - 1| + 1) - 1) and g(x) = 2(3sin(1π(2x - 4) - 2)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-1(1sin(2π(-2|x| + 2) - 1) - 3)^2 - 3(1sin(2π(-2|x| + 2) - 1) - 3) - 3) - 1)/(2(-1(1sin(2π(-2|x| + 2) - 1) - 3)^2 - 3(1sin(2π(-2|x| + 2) - 1) - 3) - 3)) and g(x) = (3(2sin(1π(5x^2 - 1) - 1) + 1))/(-3(2sin(1π(5x^2 - 1) - 1)... | 230 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4((-1(3e^(-1(-3x^2 + 1))) - 3)/(-1(3e^(-1(-3x^2 + 1))) - 2))^2 - 3((-1(3e^(-1(-3x^2 + 1))) - 3)/(-1(3e^(-1(-3x^2 + 1))) - 2)) - 2 and g(x) = (3((-3(-2sin(2πx - 1) - 2))/(-1(-2sin(2πx - 1) - 2) - 1)) + 3)/(1((-3(-2sin(2πx - 1) - 2))/(-1(-2sin(2πx - 1) -... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((-3(-5(3cos(1πx) + 2)) - 3)/(-1(-5(3cos(1πx) + 2)) + 2)) + 2| - 2 and g(x) = 1(5(2x^2 - 5x - 1) - 3) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-3|(1|(-1e^(-1x) - 1)| - 3)| - 1) - 2 and g(x) = (2(-1cos(3π(1x^2 - 1x + 3) + 3) - 1) - 2)/(-2(-1cos(3π(1x^2 - 1x + 3) + 3) - 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 174 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(3(1(1cos(1πx + 3) - 1)^2 + 4) - 2) + 3)/(2(3(1(1cos(1πx + 3) - 1)^2 + 4) - 2) - 1) and g(x) = 3|(3cos(3π(2|x + 2| + 1) - 1) - 2)| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 240 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2|(-3e^(2(-3|x + 2| - 2)) - 2) + 2| - 3) and g(x) = 5(1cos(1π(2x + 1) + 1))^2 + 1(1cos(1π(2x + 1) + 1)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(2(5(3cos(1πx - 3) + 2) - 5)^2 - 3(5(3cos(1πx - 3) + 2) - 5) - 5) + 3| + 2 and g(x) = -2((-2(-1cos(1πx + 2) + 1) + 2)/(2(-1cos(1πx + 2) + 1) - 1)) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your... | 66 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-3|(1|((2x)/(-2x - 3)) + 3| + 2) - 3| - 1) - 1)/(2(-3|(1|((2x)/(-2x - 3)) + 3| + 2) - 3| - 1)) and g(x) = 3cos(3π((3(-2x^2 - 2x + 2) - 3)/(2(-2x^2 - 2x + 2) - 3)) + 3) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex... | 51 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(2sin(2π(2(-5x^2 + 3x - 4)^2 - 3(-5x^2 + 3x - 4) - 4) - 2) + 3) - 1 and g(x) = -4(-3e^(-2(-2x - 5)) - 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 377 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(1(-3sin(3π(1|x - 2|) + 1) - 1)^2 - 2))/(1(1(-3sin(3π(1|x - 2|) + 1) - 1)^2 - 2) + 1) and g(x) = -1|(1|(5x^2 + 2x - 5) + 1| - 3) - 2| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 21 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(-5(-2cos(2π((2x - 2)/(3x - 2))) - 2) + 1)^2 - 3(-5(-2cos(2π((2x - 2)/(3x - 2))) - 2) + 1) + 1 and g(x) = -2cos(1π(-4(1x^2 - 2x + 2) - 4) - 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(2|(-1(-4x + 2) - 1) + 1| - 1) + 4 and g(x) = 3(-1cos(1π(4x^2 - 2x) + 3) - 1) + 5 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) = 1(3|(2(-4x^2 + 1x + 2)^2 - 3(-4x^2 + 1x + 2) + 1) - 1| + 3) - 5 and g(x) = 1|(-1|(1x^2 + 3x) + 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) = -2(1(-2cos(2π(4x^2 + 5x + 1) - 2) - 2)^2 + 1(-2cos(2π(4x^2 + 5x + 1) - 2) - 2) - 1)^2 - 1(1(-2cos(2π(4x^2 + 5x + 1) - 2) - 2)^2 + 1(-2cos(2π(4x^2 + 5x + 1) - 2) - 2) - 1) - 2 and g(x) = -4(1|(-5x + 5) + 2| + 2) + 2 in the range -10... | 989 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(2π(4(-3((-1x - 2)/(-3x))^2 - 3((-1x - 2)/(-3x)) - 2)^2 + 4)) - 2 and g(x) = (3(5(-2x - 3) + 4) - 3)/(-3(5(-2x - 3) + 4)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 376 | MATH |
[
{
"content": "How many solutions does the equation (2((-1((-2(-3x^2 + 1x - 5) - 2)/(2(-3x^2 + 1x - 5) - 3)) - 1)/(-3((-2(-3x^2 + 1x - 5) - 2)/(2(-3x^2 + 1x - 5) - 3)) - 1)) + 2)/(3((-1((-2(-3x^2 + 1x - 5) - 2)/(2(-3x^2 + 1x - 5) - 3)) - 1)/(-3((-2(-3x^2 + 1x - 5) - 2)/(2(-3x^2 + 1x - 5) - 3)) - 1))) = 2(-2|(3e^... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-3|(-3(2sin(3πx - 3) - 3)^2 - 1(2sin(3πx - 3) - 3)) + 3| + 1)^2 - 5(-3|(-3(2sin(3πx - 3) - 3)^2 - 1(2sin(3πx - 3) - 3)) + 3| + 1) + 4 and g(x) = -1sin(3π(-1(2x - 4) + 4) + 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer ... | 220 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(3(-2|(4x - 1) - 2| + 3)^2 - 4(-2|(4x - 1) - 2| + 3) + 2) - 2)/(2(3(-2|(4x - 1) - 2| + 3)^2 - 4(-2|(4x - 1) - 2| + 3) + 2) - 1) and g(x) = 2|(1cos(2π(2x^2 + 2x) - 1) + 2) - 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in La... | 721 | MATH |
[
{
"content": "How many solutions does the equation 2(1cos(1π(-5(-1|x - 2| - 2)^2 - 4(-1|x - 2| - 2) - 5) - 2) - 3)^2 - 3(1cos(1π(-5(-1|x - 2| - 2)^2 - 4(-1|x - 2| - 2) - 5) - 2) - 3) - 3 = 2|(2|(1cos(1πx + 2)) + 3| + 2) - 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxe... | 674 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(2π((-3(-4(1x^2 - 2x) - 5) + 3)/(3(-4(1x^2 - 2x) - 5) + 3)) - 2) + 3 and g(x) = 4(1(1x^2 - 3x)^2 + 5(1x^2 - 3x))^2 - 3(1(1x^2 - 3x)^2 + 5(1x^2 - 3x)) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1((-2(1e^(-1(-1|x - 1| + 2)) + 3))/(1(1e^(-1(-1|x - 1| + 2)) + 3) + 2)) + 3 = 2|((-2(-3sin(2πx + 2) - 2) - 3)/(-2(-3sin(2πx + 2) - 2) + 3)) + 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((-1(-1(-2|x - 2| - 2) - 1) + 1)/(2(-1(-2|x - 2| - 2) - 1) + 1)) - 4 and g(x) = 3sin(2π(3|(-2x - 4) - 3| + 3) - 2) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 70 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-1|((-2(-5x^2 - 1x + 5) + 2)/(-1(-5x^2 - 1x + 5) + 3)) + 1| - 1) - 4 and g(x) = -1(-5((-1x + 2)/(-1x)) - 5)^2 - 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) = (3(-1e^(-1(1|(-1x + 1)| + 1)) + 1) - 3)/(2(-1e^(-1(1|(-1x + 1)| + 1)) + 1)) and g(x) = (2(4(-5x^2 - 4x + 1)^2 - 2(-5x^2 - 4x + 1) - 1) - 2)/(1(4(-5x^2 - 4x + 1)^2 - 2(-5x^2 - 4x + 1) - 1) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent ... | 4 | MATH |
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