messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation -5(-3sin(3π(5(-2|x - 2| - 3) + 3)) - 3)^2 - 4(-3sin(3π(5(-2|x - 2| - 3) + 3)) - 3) - 2 = -2(2e^(2((1x - 2)/(2x + 2)))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 492 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(1((-3(-2sin(1πx + 3) - 1) - 3)/(-2(-2sin(1πx + 3) - 1) - 1))^2 - 5((-3(-2sin(1πx + 3) - 1) - 3)/(-2(-2sin(1πx + 3) - 1) - 1)) + 2) - 4 and g(x) = (-2(3|(3|x - 2| + 1) + 1| + 1) + 2)/(-3(3|(3|x - 2| + 1) + 1| + 1) - 3) intersect in the interval [-10, ... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-1|(3((-2x + 2)/(-1x - 1)) + 4) - 1| + 3)^2 - 1(-1|(3((-2x + 2)/(-1x - 1)) + 4) - 1| + 3) + 1 and g(x) = (1(-1|((-3x - 2)/(-2x - 3)) + 3| + 1) - 2)/(1(-1|((-3x - 2)/(-2x - 3)) + 3| + 1) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-2(-4((-3x)/(2x - 1)) - 2)^2 - 2(-4((-3x)/(2x - 1)) - 2) + 3)^2 - 3(-2(-4((-3x)/(2x - 1)) - 2)^2 - 2(-4((-3x)/(2x - 1)) - 2) + 3) + 1 and g(x) = 5(3cos(3π(-4x + 2) - 3) - 3)^2 + 3(3cos(3π(-4x + 2) - 3) - 3) - 5 in the range -10 ≤... | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(4(-5(2x^2 - 4) + 2) + 4)^2 - 4(4(-5(2x^2 - 4) + 2) + 4) + 3 and g(x) = (3(-1e^(-1(-1|x + 3| - 2))) + 1)/(2(-1e^(-1(-1|x + 3| - 2))) - 1) in the range -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) = 2(3|((1(-1x^2 + 1x + 1) + 2)/(2(-1x^2 + 1x + 1) + 1)) - 1| - 2)^2 - 2(3|((1(-1x^2 + 1x + 1) + 2)/(2(-1x^2 + 1x + 1) + 1)) - 1| - 2) - 1 and g(x) = (-2(3|(1|x - 2| + 2)| + 2) + 3)/(-1(3|(1|x - 2| + 2)| + 2) + 2) in the range -10 ≤ x... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-3sin(3π(2((1x - 2)/(-2x - 2)) + 5) - 2))^2 - 3 and g(x) = -2(1sin(3π(1|x + 3| - 1))) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 104 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(1|(3((1x + 3)/(-1x + 3))^2 + 1((1x + 3)/(-1x + 3)) + 2) - 3| + 1) - 1 and g(x) = -2(-2e^(-1(-2x + 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 solutions does the equation (-1((2(-1|(3x^2 + 5x + 2) + 2| + 1) - 3)/(1(-1|(3x^2 + 5x + 2) + 2| + 1))) - 1)/(2((2(-1|(3x^2 + 5x + 2) + 2| + 1) - 3)/(1(-1|(3x^2 + 5x + 2) + 2| + 1))) - 3) = 2(3|(-1cos(3πx) + 3) - 1| - 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in La... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(3(-3(5x^2 + 4x - 2) - 1) - 5))/(-3(3(-3(5x^2 + 4x - 2) - 1) - 5) - 2) and g(x) = 2cos(2π(3(2x - 1) - 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 242 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-1sin(3π((-1(4x^2 + 5x - 2) - 2)/(1(4x^2 + 5x - 2))) - 3) + 3) + 2)/(1(-1sin(3π((-1(4x^2 + 5x - 2) - 2)/(1(4x^2 + 5x - 2))) - 3) + 3) - 2) and g(x) = 1(2(2cos(2πx + 2) - 1)^2 - 1(2cos(2πx + 2) - 1) - 2) + 2 in the range -10 ≤ x ... | 66 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5((1((2(2cos(2πx - 3) + 3))/(-1(2cos(2πx - 3) + 3) - 3)) + 3)/(2((2(2cos(2πx - 3) + 3))/(-1(2cos(2πx - 3) + 3) - 3)) + 3))^2 + 4((1((2(2cos(2πx - 3) + 3))/(-1(2cos(2πx - 3) + 3) - 3)) + 3)/(2((2(2cos(2πx - 3) + 3))/(-1(2cos(2πx - ... | 39 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-3cos(2π(1|(-3|x - 2| - 1) - 2| - 1) - 1) - 2) + 2 and g(x) = -2e^(-2(-3((-3x + 3)/(-1x - 2)) + 4)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(5(-2cos(3π(2x + 3) - 3) + 2)^2 + 3(-2cos(3π(2x + 3) - 3) + 2) - 1)^2 + 5(5(-2cos(3π(2x + 3) - 3) + 2)^2 + 3(-2cos(3π(2x + 3) - 3) + 2) - 1) - 5 and g(x) = -4(2(2cos(1πx) + 2) - 4) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex fo... | 80 | MATH |
[
{
"content": "How many solutions does the equation 3|(-2(-2|(1|x - 1| - 2) - 2|)^2 - 4(-2|(1|x - 1| - 2) - 2|)) - 1| - 1 = (2(3cos(3π((3x)/(3x + 1)) + 1) + 2) + 1)/(-3(3cos(3π((3x)/(3x + 1)) + 1) + 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(2π((-2(2|((-1x)/(2x))| - 1) + 3)/(1(2|((-1x)/(2x))| - 1) - 3)) - 3) - 1 and g(x) = -5(1(-5x - 2)^2 + 4(-5x - 2) - 5)^2 - 2(1(-5x - 2)^2 + 4(-5x - 2) - 5) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-2(-2cos(1π(-5x^2 + 2x + 1)) + 1) + 3) - 3| + 2 and g(x) = 2|((-2(-5x^2 + 1x - 5) + 3)/(-1(-5x^2 + 1x - 5) + 2)) - 2| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 37 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(1(2((-1x - 2)/(3x + 1))^2 - 4)^2 - 4(2((-1x - 2)/(3x + 1))^2 - 4) - 2) + 1)/(-3(1(2((-1x - 2)/(3x + 1))^2 - 4)^2 - 4(2((-1x - 2)/(3x + 1))^2 - 4) - 2)) and g(x) = (2(-2|(-2|x + 1| + 3) + 3| + 1) - 3)/(1(-2|(-2|x + 1| + 3) + 3| + 1) + 1) intersect in... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(3π(-3(2|((-1x - 1)/(2x + 3)) - 1| + 3)^2 - 4) - 2) - 1 and g(x) = 5(-1((-2x + 1)/(2x - 3))^2 + 3((-2x + 1)/(2x - 3)) + 1) 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|(-2|((2((-1x - 3)/(-3x - 2)) - 3)/(-3((-1x - 3)/(-3x - 2)) + 4)) - 2| + 1) + 2| + 2 = 5(1cos(2π(4x^2 - 2) - 2) + 2) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1348 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-2(-4(-1e^(1x) + 3) + 4) + 1)/(-1(-4(-1e^(1x) + 3) + 4))) - 2 and g(x) = -2(-1sin(2π(2|x - 1| - 3) + 2))^2 + 4(-1sin(2π(2|x - 1| - 3) + 2)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 79 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2(1(1x - 4) + 3)^2 + 3(1(1x - 4) + 3) - 1) + 1)/(2(2(1(1x - 4) + 3)^2 + 3(1(1x - 4) + 3) - 1) + 1) and g(x) = 1sin(3π(3|(1x^2 + 3x) + 2| - 2)) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 1363 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((-1((-2(3sin(2πx + 1)) + 1)/(1(3sin(2πx + 1)) - 2)) + 1)/(-1((-2(3sin(2πx + 1)) + 1)/(1(3sin(2πx + 1)) - 2)) - 2)) + 1)/(-1((-1((-2(3sin(2πx + 1)) + 1)/(1(3sin(2πx + 1)) - 2)) + 1)/(-1((-2(3sin(2πx + 1)) + 1)/(1(3sin(2πx + 1)) -... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(2|(-3(-5x^2 + 4x + 2) + 3) - 2| + 1) - 3 and g(x) = -2(-1sin(2π(-2x^2 - 5x + 2) - 2))^2 + 1(-1sin(2π(-2x^2 - 5x + 2) - 2)) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 10 | MATH |
[
{
"content": "How many solutions does the equation 2|(5(-1(3x - 1)^2 - 4(3x - 1) + 4) + 3) - 3| - 1 = -2cos(1π(5(-4x^2 + 2)^2 - 1(-4x^2 + 2) + 2) + 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 (-2(-2((3(-1x^2 + 5x - 5) - 3)/(2(-1x^2 + 5x - 5) + 1))^2 + 3((3(-1x^2 + 5x - 5) - 3)/(2(-1x^2 + 5x - 5) + 1)) - 5))/(3(-2((3(-1x^2 + 5x - 5) - 3)/(2(-1x^2 + 5x - 5) + 1))^2 + 3((3(-1x^2 + 5x - 5) - 3)/(2(-1x^2 + 5x - 5) + 1)) - 5) - 2) = 4(-3cos(1π(2|x - 3| - ... | 80 | MATH |
[
{
"content": "How many solutions does the equation 3cos(2π(-2(-5(2x^2 - 2)^2 - 1(2x^2 - 2) + 1) + 2) - 1) - 1 = (-2(-3|(2cos(1πx - 3) + 2)| - 1))/(2(-3|(2cos(1πx - 3) + 2)| - 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1183 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-2|(-3sin(1π(-2x - 2) + 1) - 3) + 2| + 2))/(-1(-2|(-3sin(1π(-2x - 2) + 1) - 3) + 2| + 2) + 2) and g(x) = 3sin(3π((1(-2|x - 3| + 2) - 1)/(1(-2|x - 3| + 2) + 1)) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\b... | 136 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2((1(-1|(1x - 1) + 3| + 1) + 2)/(-2(-1|(1x - 1) + 3| + 1) + 3)))/(1((1(-1|(1x - 1) + 3| + 1) + 2)/(-2(-1|(1x - 1) + 3| + 1) + 3))) and g(x) = -3((2(3cos(3πx - 1) + 2) - 2)/(-1(3cos(3πx - 1) + 2)))^2 + 3((2(3cos(3πx - 1) + 2) - 2)/(-1(3cos(3πx - 1) + 2... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(2(-3(2x^2 - 3x)^2 - 3(2x^2 - 3x) + 3) - 3) - 2| - 3 and g(x) = 2|(3sin(3π(-5x^2 + 1x - 3) - 2) - 3) + 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 54 | MATH |
[
{
"content": "How many solutions does the equation -4((2(2(-3|x + 1|)^2 + 4(-3|x + 1|) - 2) - 2)/(-1(2(-3|x + 1|)^2 + 4(-3|x + 1|) - 2) + 2))^2 + 5((2(2(-3|x + 1|)^2 + 4(-3|x + 1|) - 2) - 2)/(-1(2(-3|x + 1|)^2 + 4(-3|x + 1|) - 2) + 2)) + 5 = 2(2|(2sin(1πx + 2) - 3) + 2| - 3) + 1 have for x in the interval [-10,... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(-5(-4((1x + 1)/(-2x))^2 - 3((1x + 1)/(-2x)) - 1) + 1)^2 + 2 = -3(-2(-1x^2 + 4)^2 + 2(-1x^2 + 4)) + 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(2((-2(1|x| + 1) - 1)/(-1(1|x| + 1) + 3))^2 + 4((-2(1|x| + 1) - 1)/(-1(1|x| + 1) + 3)) - 3)^2 - 1(2((-2(1|x| + 1) - 1)/(-1(1|x| + 1) + 3))^2 + 4((-2(1|x| + 1) - 1)/(-1(1|x| + 1) + 3)) - 3) + 4 and g(x) = 2e^(-1(-3((-3x - 2)/... | 5 | MATH |
[
{
"content": "How many solutions does the equation (-1(2sin(2π(1|(1|x + 1| + 1) + 2| - 1) - 1) + 1) + 3)/(1(2sin(2π(1|(1|x + 1| + 1) + 2| - 1) - 1) + 1) + 2) = -2(-2sin(3π(-1x + 5) + 3) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(3sin(2π(-1(-5x^2 + 1x + 3)^2 + 5(-5x^2 + 1x + 3) + 3)) + 2)^2 - 4(3sin(2π(-1(-5x^2 + 1x + 3)^2 + 5(-5x^2 + 1x + 3) + 3)) + 2) + 5 and g(x) = -1cos(2π((-2(-4x + 3) + 3)/(-2(-4x + 3) + 1)) + 1) + 2 in the range -10 ≤ x ≤ 10?\n\nPre... | 325 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(3π(3|(1(-2x^2 - 5x + 2)^2 - 3(-2x^2 - 5x + 2) - 3) + 3| + 1) + 2) + 2 and g(x) = (1(1(-2x + 1) - 4) - 1)/(-3(1(-2x + 1) - 4) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 679 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(4((1((-1x + 1)/(1x)) - 2)/(-3((-1x + 1)/(1x)) - 1))^2 - 2((1((-1x + 1)/(1x)) - 2)/(-3((-1x + 1)/(1x)) - 1)) - 5) - 3| - 1 and g(x) = -2|(-3|(3cos(3πx - 2) - 2) + 2| + 2) + 1| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer ... | 21 | MATH |
[
{
"content": "How many solutions does the equation 3|(-2|(4(-2|x - 2| + 1)) - 2| + 1) + 2| = 4(-5(-2|x + 2| + 2)^2 + 4(-2|x + 2| + 2) + 1)^2 - 5(-5(-2|x + 2| + 2)^2 + 4(-2|x + 2| + 2) + 1) - 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|((3((-1(1cos(2πx - 3) + 3) - 2)/(-2(1cos(2πx - 3) + 3) + 3)) - 3)/(-1((-1(1cos(2πx - 3) + 3) - 2)/(-2(1cos(2πx - 3) + 3) + 3)) - 3)) + 1| - 3 and g(x) = -2cos(2π((-3((-2x + 2)/(-3x + 1)) - 1)/(1((-2x + 2)/(-3x + 1)) + 1)) ... | 19 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(2π(-2(3(-3|x - 2| - 1)^2 + 1(-3|x - 2| - 1) + 4) + 3) + 3) + 1 and g(x) = (1(1((1x - 2)/(1x - 3))^2 - 5((1x - 2)/(1x - 3)) + 3) + 1)/(3(1((1x - 2)/(1x - 3))^2 - 5((1x - 2)/(1x - 3)) + 3) + 1) intersect in the interval [-10, 10].\n\nPresent the an... | 1419 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4((-2(3(-3|x + 1| + 2) + 1) + 3)/(3(3(-3|x + 1| + 2) + 1) + 1))^2 - 5((-2(3(-3|x + 1| + 2) + 1) + 3)/(3(3(-3|x + 1| + 2) + 1) + 1)) - 3 and g(x) = 2sin(1π(-5(-5x + 2)) - 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaT... | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(1π(2|(5(-1x^2 + 3x - 3)) + 3| - 2) + 2) + 3 and g(x) = 2|((-2(1x^2 + 4x - 1))/(-3(1x^2 + 4x - 1) + 4)) - 1| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 134 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3|(1sin(2π(-3|x| + 1) - 2) - 3) + 3| - 1) + 2 and g(x) = 1(3|(1x) + 2| - 2)^2 - 5(3|(1x) + 2| - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(3π((1(5(-3|x + 1| + 2) - 5) + 3)/(1(5(-3|x + 1| + 2) - 5) + 1)) - 3) + 2 and g(x) = -1e^(1(1(-4x^2 + 1x + 2)^2 - 4(-4x^2 + 1x + 2) + 3)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(3(3(-1x - 5) - 2) + 5) + 1 and g(x) = -4(2(3sin(3πx) - 3)^2 + 2(3sin(3πx) - 3))^2 + 2(2(3sin(3πx) - 3)^2 + 2(3sin(3πx) - 3)) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 46 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-2((-3(1sin(3πx - 1) + 2) - 3)/(-2(1sin(3πx - 1) + 2))) + 3)/(2((-3(1sin(3πx - 1) + 2) - 3)/(-2(1sin(3πx - 1) + 2))) + 2))^2 - 4((-2((-3(1sin(3πx - 1) + 2) - 3)/(-2(1sin(3πx - 1) + 2))) + 3)/(2((-3(1sin(3πx - 1) + 2) - 3)/(-2(1s... | 47 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((3(-1|(5x + 3) - 1| + 3) + 3)/(2(-1|(5x + 3) - 1| + 3) - 1)) - 5 and g(x) = (-1(-3|((3x + 2)/(-1x + 2)) + 3| + 2) - 2)/(1(-3|((3x + 2)/(-1x + 2)) + 3| + 2) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\b... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-2|(-1(2x^2 + 4x + 3)^2 + 3) + 2| + 1)^2 + 3(-2|(-1(2x^2 + 4x + 3)^2 + 3) + 2| + 1) - 3 and g(x) = -1|((-2(-3|x + 1| - 2) + 2)/(-1(-3|x + 1| - 2) - 1)) + 1| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-2|(-2sin(1π(5x^2 + 4x + 4) + 2) - 1)| - 1) - 1 and g(x) = -3(3(-2e^(-1x) - 1)^2 - 4(-2e^(-1x) - 1) - 4)^2 + 3(3(-2e^(-1x) - 1)^2 - 4(-2e^(-1x) - 1) - 4) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\box... | 427 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(2|(1(3sin(1πx - 2) + 1)^2 + 5(3sin(1πx - 2) + 1) - 4) - 2| - 2)^2 + 2(2|(1(3sin(1πx - 2) + 1)^2 + 5(3sin(1πx - 2) + 1) - 4) - 2| - 2) and g(x) = 3sin(1π(3(-2|x - 3| - 1)^2 - 4(-2|x - 3| - 1) + 4) - 1) where -10 ≤ x ≤ 10.\n... | 46 | MATH |
[
{
"content": "How many solutions does the equation (2(-3(3cos(3π(-3x + 1) - 1) + 2) - 1) + 3)/(3(-3(3cos(3π(-3x + 1) - 1) + 2) - 1) + 2) = 3(4((-3x - 3)/(-3x + 1))^2 + 1((-3x - 3)/(-3x + 1)) + 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((-2(-2|(-2|x|) + 3| - 3) + 1)/(3(-2|(-2|x|) + 3| - 3) + 1)))/(3((-2(-2|(-2|x|) + 3| - 3) + 1)/(3(-2|(-2|x|) + 3| - 3) + 1)) - 2) and g(x) = 5(3sin(1π(3|x - 1| + 1) - 3)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in La... | 62 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(5(-1(-3x - 4)^2 + 1(-3x - 4)) + 1)^2 + 4(5(-1(-3x - 4)^2 + 1(-3x - 4)) + 1) - 2 and g(x) = (-1(-1e^(2(-4x + 3)) + 1) - 1)/(3(-1e^(2(-4x + 3)) + 1) - 1) intersect in the interval [-10, 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) = 1|((-3(-3cos(3π((-1x - 2)/(-2x + 1)) - 1) - 1) + 3)/(1(-3cos(3π((-1x - 2)/(-2x + 1)) - 1) - 1))) - 3| + 3 and g(x) = (3((-1((2x + 3)/(-2x + 3)) + 3)/(2((2x + 3)/(-2x + 3)) + 1)) - 3)/(-2((-1((2x + 3)/(-2x + 3)) + 3)/(2((2x + 3)/(-2... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-2cos(1π(4(3|x + 2| + 3)) - 3) - 3) + 5 and g(x) = (1(1(3x^2 - 5x + 2) + 2) + 3)/(-1(1(3x^2 - 5x + 2) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 240 | MATH |
[
{
"content": "How many solutions does the equation -1((2(-1sin(3π(-3x + 1) + 1) + 1) - 1)/(3(-1sin(3π(-3x + 1) + 1) + 1)))^2 - 1((2(-1sin(3π(-3x + 1) + 1) + 1) - 1)/(3(-1sin(3π(-3x + 1) + 1) + 1))) + 3 = -3cos(2π(2((-3x - 2)/(3x - 1)) + 5) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in ... | 179 | MATH |
[
{
"content": "How many solutions does the equation 3(-3|(-2sin(1π(5x^2 - 1x - 4)) - 3) + 3|) - 5 = (-1(-2cos(1π(-4x^2 - 1x + 4) + 2)) - 2)/(-2(-2cos(1π(-4x^2 - 1x + 4) + 2)) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 106 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-2sin(2π(5((1x)/(-1x - 2))^2 + 1((1x)/(-1x - 2)) + 1) - 3)) - 2 and g(x) = -1(-3(1x^2 - 2x - 4) - 3)^2 - 4(-3(1x^2 - 2x - 4) - 3) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 31 | MATH |
[
{
"content": "How many solutions does the equation 2|(-1((2(-1x^2 + 5x - 5) + 1)/(1(-1x^2 + 5x - 5) - 3)) + 2) + 1| - 2 = -3|(1(-3|x + 2| + 2)) + 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": "How many points of intersection exist between the functions f(x) = 3(2sin(2π(-4(-4x^2 + 2x - 2)^2 - 5(-4x^2 + 2x - 2) + 4) + 2))^2 + 4(2sin(2π(-4(-4x^2 + 2x - 2)^2 - 5(-4x^2 + 2x - 2) + 4) + 2)) + 4 and g(x) = (-2(-3|((-2x + 1)/(2x - 3)) + 2| + 2))/(3(-3|((-2x + 1)/(2x - 3)) + 2| + 2) + 2) in the ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(1((-2(2e^(1x) - 3) + 3)/(-2(2e^(1x) - 3) + 1))^2 + 3((-2(2e^(1x) - 3) + 3)/(-2(2e^(1x) - 3) + 1))) - 3 and g(x) = 5(3|(3x) + 1|)^2 + 1(3|(3x) + 1|) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "How many solutions does the equation (2((1(-1sin(1π(-2|x + 1| + 3) + 1)) - 2)/(2(-1sin(1π(-2|x + 1| + 3) + 1)) - 3)) - 1)/(3((1(-1sin(1π(-2|x + 1| + 3) + 1)) - 2)/(2(-1sin(1π(-2|x + 1| + 3) + 1)) - 3))) = (-2(-3cos(3π(1|x - 2| - 1) + 3) + 1) + 3)/(-3(-3cos(3π(1|x - 2| - 1) + 3) + 1) + 4) have for ... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-5(5((3x + 3)/(-2x + 3)) + 4) + 5) + 1 and g(x) = (-2(-2cos(2π(-1|x - 1| + 1) + 1) + 2) - 2)/(-3(-2cos(2π(-1|x - 1| + 1) + 1) + 2) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 14 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(4(-1|((-1x + 3)/(-1x - 2)) + 3|) - 5)| - 1 and g(x) = (1(-3(3cos(3πx - 2) + 3) + 2) - 2)/(2(-3(3cos(3πx - 2) + 3) + 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "How many solutions does the equation -1sin(1π(1(-4(-1|x - 2| + 1)^2 - 4(-1|x - 2| + 1) + 5)^2 + 1(-4(-1|x - 2| + 1)^2 - 4(-1|x - 2| + 1) + 5)) - 1) - 1 = -3cos(1π(4((1x + 3)/(3x))^2 - 3((1x + 3)/(3x)) + 2) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxe... | 107 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-2((-2(5x^2 + 1x - 2) - 1)/(-2(5x^2 + 1x - 2) - 2)) + 3) + 3| - 2 and g(x) = -1|((-3(1x - 3) - 1)/(2(1x - 3) - 1)) - 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-3(-1(1x^2 - 5x + 3) - 2)^2 + 1)^2 - 3(-3(-1(1x^2 - 5x + 3) - 2)^2 + 1) - 4 and g(x) = (-1(-1((-1x + 1)/(-2x))^2 + 4((-1x + 1)/(-2x))) - 2)/(2(-1((-1x + 1)/(-2x))^2 + 4((-1x + 1)/(-2x))) - 1) intersect in the interval [-10, 10].\n\nPresent the answer... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(3π(5((-3(-1x - 5))/(-2(-1x - 5) - 2)) - 1) + 2) + 2 and g(x) = -4(1(-3cos(1πx + 3) - 1) + 1) + 1 in the range -10 ≤ x ≤ 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) = -2((1(1(5x^2) + 4) - 3)/(3(1(5x^2) + 4) - 2))^2 and g(x) = 3(-2|(2|x + 2|) - 2|) 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 -1(1cos(2π(3|(-2x^2 + 2) + 1| - 2) - 2) - 3) - 2 = (3(-2cos(2π((3x)/(1x - 2))) + 2) + 1)/(-1(-2cos(2π((3x)/(1x - 2))) + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 502 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-3sin(2π((2(2x + 1) + 3)/(1(2x + 1) - 3)) - 3) + 2) + 1)/(3(-3sin(2π((2(2x + 1) + 3)/(1(2x + 1) - 3)) - 3) + 2) - 2) and g(x) = 4(-1((-3x + 1)/(3x + 2)) - 5) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{You... | 21 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3((1(-1|(5x - 3) + 2| + 1) + 3)/(-1(-1|(5x - 3) + 2| + 1) - 1)) + 2)/(-2((1(-1|(5x - 3) + 2| + 1) + 3)/(-1(-1|(5x - 3) + 2| + 1) - 1)) - 2) and g(x) = -3(2sin(2π(-2|x - 2|) - 2) - 3) + 5 where -10 ≤ x ≤ 10.\n\nPresent the an... | 16 | MATH |
[
{
"content": "How many solutions does the equation -1e^(1(-3(4(1x^2 + 5x + 5)^2 - 3(1x^2 + 5x + 5) - 4) - 1)) + 2 = -1(5(3e^(1x) + 2))^2 - 3(5(3e^(1x) + 2)) - 4 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(-2(-2cos(2π(-1|x - 3| - 1) - 3))^2 + 4(-2cos(2π(-1|x - 3| - 1) - 3)) + 1) + 5 and g(x) = 2(1|(3x^2 + 1x - 2) - 2| - 3) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π(-2|(-2|(5x + 3) - 2| + 1) + 3| - 3)) + 2 and g(x) = -1(-3cos(1π(-2x^2 + 5x - 3) + 2) - 3) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 265 | MATH |
[
{
"content": "How many solutions does the equation -3e^(2(2|((1((-1x - 3)/(3x - 2)) + 3)/(3((-1x - 3)/(3x - 2)) + 2))| - 1)) + 1 = -2cos(2π(-4(1|x + 2| - 1)^2 - 4(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"
... | 19 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(1π(-2|(5((2x)/(1x - 3))^2 - 1((2x)/(1x - 3))) - 1| - 2) + 1) - 3 and g(x) = (-2(-2|(-1|x + 1| - 3) + 3| + 3) + 2)/(3(-2|(-1|x + 1| - 3) + 3| + 3)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 29 | MATH |
[
{
"content": "How many solutions does the equation 2sin(1π(2(1|(-1x - 2) + 1| + 1)^2 - 5(1|(-1x - 2) + 1| + 1) - 4) + 1) - 2 = 1(2sin(2π(-4x - 2) + 1) + 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 285 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(3(-2sin(1π((1x - 2)/(1x - 3))) + 2)^2 - 5(-2sin(1π((1x - 2)/(1x - 3))) + 2) + 1) + 1 and g(x) = -5(-3|(2x^2 + 2x - 2) - 1| + 3)^2 + 1(-3|(2x^2 + 2x - 2) - 1| + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed... | 4 | MATH |
[
{
"content": "How many solutions does the equation (3(3(3cos(2π(-1x + 1) + 2) + 1)^2 + 4(3cos(2π(-1x + 1) + 2) + 1) - 2) + 2)/(-3(3(3cos(2π(-1x + 1) + 2) + 1)^2 + 4(3cos(2π(-1x + 1) + 2) + 1) - 2) + 1) = 4((-3(1|x - 2| - 1) - 3)/(1(1|x - 2| - 1))) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer ... | 24 | MATH |
[
{
"content": "How many solutions does the equation (2((3((-2(2cos(3πx - 3) - 1) - 3)/(-3(2cos(3πx - 3) - 1) + 4)) - 2)/(3((-2(2cos(3πx - 3) - 1) - 3)/(-3(2cos(3πx - 3) - 1) + 4)) + 1)) - 1)/(1((3((-2(2cos(3πx - 3) - 1) - 3)/(-3(2cos(3πx - 3) - 1) + 4)) - 2)/(3((-2(2cos(3πx - 3) - 1) - 3)/(-3(2cos(3πx - 3) - 1) ... | 39 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(1π(-4(-1|(-1x^2 - 5x - 4) - 2| - 3) - 4) + 1) - 2 and g(x) = -3(3e^(1(-4x - 3)) - 1)^2 + 3(3e^(1(-4x - 3)) - 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 600 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(3π(2|(3|((2x - 2)/(-1x - 2)) - 1| - 3) + 3| - 2)) - 1 and g(x) = -4(-1cos(3π(-3x^2 + 1) - 2)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1396 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-3(2|(-1|x - 3|) + 3| - 3) + 3) - 1 and g(x) = 2|(3cos(1π(-1x^2 + 1x + 2) + 1) + 2)| + 1 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) = -1|(-3|((-3((3x - 2)/(1x + 2)) + 2)/(3((3x - 2)/(1x + 2)) + 1)) - 1| - 1) + 3| + 1 and g(x) = (1(3(-1x^2 + 5x - 1) - 1) - 2)/(2(3(-1x^2 + 5x - 1) - 1) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|((-2(-2(-1|x + 3| + 3) - 1) + 3)/(-2(-2(-1|x + 3| + 3) - 1))) + 1| + 3 and g(x) = 1(2|(-4x + 2) - 3| - 1) - 4 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(3((1x - 2)/(-1x - 2)) + 2))/(-2(3((1x - 2)/(-1x - 2)) + 2) - 3)) + 2)/(1((-1(3((1x - 2)/(-1x - 2)) + 2))/(-2(3((1x - 2)/(-1x - 2)) + 2) - 3))) = 1cos(2π(-1(-3|x + 2|)) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format:... | 25 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-3(3|((-3x - 1)/(-3x - 2)) + 1| - 2)^2 + 3(3|((-3x - 1)/(-3x - 2)) + 1| - 2) - 1) + 5 and g(x) = -3e^(-1(2(1x^2 - 4x - 2) + 5)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-1|(-3(-1x - 2)^2 - 4(-1x - 2) + 4) - 3| + 2) + 1 and g(x) = (2(2sin(3π(-5x^2 - 3x + 5)) + 3) + 2)/(1(2sin(3π(-5x^2 - 3x + 5)) + 3) + 2) where -10 ≤ x ≤ 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) = -1sin(2π(3(-3(-1x^2 + 5) - 3)^2 - 5(-3(-1x^2 + 5) - 3) - 4) - 3) + 1 and g(x) = -1|(3sin(1π((1x - 1)/(3x - 1)) + 3) + 1) + 2| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 223 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(3π(5(-1(4x^2 - 5x + 2)^2 + 4(4x^2 - 5x + 2) + 4)^2 + 5(-1(4x^2 - 5x + 2)^2 + 4(4x^2 - 5x + 2) + 4) + 3) - 2) - 1 and g(x) = -1|(-2((2x)/(1x + 2)) - 4) - 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{You... | 60 | MATH |
[
{
"content": "How many solutions does the equation 3((2((1(-2sin(3πx + 1) + 3) - 3)/(2(-2sin(3πx + 1) + 3) - 1)) - 3)/(2((1(-2sin(3πx + 1) + 3) - 3)/(2(-2sin(3πx + 1) + 3) - 1)) - 1)) = (-3(-3e^(2((-3x - 1)/(1x))) - 2) - 1)/(-1(-3e^(2((-3x - 1)/(1x))) - 2) - 1) have for x in the interval [-10, 10]?\n\nPresent t... | 59 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(4(-2e^(2(5x^2 + 3x - 4)) + 1) - 5) + 1)/(-2(4(-2e^(2(5x^2 + 3x - 4)) + 1) - 5) + 3) and g(x) = -2(-3(-3|x - 2| + 1)^2 - 5(-3|x - 2| + 1) - 1)^2 - 2(-3(-3|x - 2| + 1)^2 - 5(-3|x - 2| + 1) - 1) + 4 where -10 ≤ x ≤ 10.\n\nPre... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-2e^(1(-4((2x - 3)/(-1x)))) - 1))/(3(-2e^(1(-4((2x - 3)/(-1x)))) - 1) - 2) and g(x) = -2sin(2π((-3(2|x + 3| - 1) - 1)/(2(2|x + 3| - 1) - 3)) + 2) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{You... | 92 | MATH |
[
{
"content": "How many solutions does the equation (-3(4(5(2|x - 2| + 1) + 5)^2 + 4(5(2|x - 2| + 1) + 5) - 3) + 1)/(-2(4(5(2|x - 2| + 1) + 5)^2 + 4(5(2|x - 2| + 1) + 5) - 3) + 3) = 5(3sin(3π(3x^2 - 1x - 2) - 3) + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 1118 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((-2(3|(-3x^2 - 2x - 3) + 3| - 3) - 1)/(1(3|(-3x^2 - 2x - 3) + 3| - 3))) + 5 and g(x) = (-1(-1|(-3x + 2) - 1| - 1) - 1)/(-3(-1|(-3x + 2) - 1| - 1) + 1) intersect in the interval [-10, 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) = (-2((-3(-3cos(1π(-3x - 2) + 3) - 1) - 3)/(3(-3cos(1π(-3x - 2) + 3) - 1) + 2)))/(2((-3(-3cos(1π(-3x - 2) + 3) - 1) - 3)/(3(-3cos(1π(-3x - 2) + 3) - 1) + 2)) + 3) and g(x) = 1cos(2π((2((-1x - 1)/(3x - 1)) - 2)/(1((-1x - 1)/(3x - 1)) ... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-1(-3|(3x - 1) - 3| - 1)^2 + 1))/(2(-1(-3|(3x - 1) - 3| - 1)^2 + 1) + 1) and g(x) = 3cos(3π(-2|((2x + 2)/(-2x)) - 1| + 1) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 91 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(-2e^(-1(-1(-2x^2 - 4x + 1) - 2))) + 1| + 1 and g(x) = (3((2(-1sin(1πx - 2) + 1) + 2)/(-3(-1sin(1πx - 2) + 1))) - 2)/(-2((2(-1sin(1πx - 2) + 1) + 2)/(-3(-1sin(1πx - 2) + 1))) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex f... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2((-1(-4(-2e^(-1x) - 2)) - 2)/(-3(-4(-2e^(-1x) - 2)))) - 1)/(-3((-1(-4(-2e^(-1x) - 2)) - 2)/(-3(-4(-2e^(-1x) - 2))))) and g(x) = -3cos(1π(1|((2x + 3)/(-1x + 2)) + 1| - 3)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaT... | 87 | MATH |
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