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) = -5((1(-1((2x - 3)/(2x)) - 4) - 3)/(-1(-1((2x - 3)/(2x)) - 4)))^2 - 1((1(-1((2x - 3)/(2x)) - 4) - 3)/(-1(-1((2x - 3)/(2x)) - 4))) + 5 and g(x) = -5(3sin(1π(1x - 2) - 2) + 1)^2 + 5(3sin(1π(1x - 2) - 2) + 1) - 3 where -10 ≤ x ≤ ... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(1((1(-2sin(3πx + 3) - 3))/(-1(-2sin(3πx + 3) - 3) + 2)) + 1) - 3)/(1(1((1(-2sin(3πx + 3) - 3))/(-1(-2sin(3πx + 3) - 3) + 2)) + 1)) and g(x) = -2|((3((3x)/(-2x - 2)) + 1)/(3((3x)/(-2x - 2)) + 2))| - 1 in the range -10 ≤ x ≤ 10?\n... | 46 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(2sin(2π(1|(3x^2 - 1x - 2) + 3| - 3) - 2) + 2) + 4 and g(x) = 4(-2cos(1π(3|x - 3| - 2) + 1) + 2)^2 + 2(-2cos(1π(3|x - 3| - 2) + 1) + 2) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 442 | MATH |
[
{
"content": "How many solutions does the equation 1sin(1π(1|(5(5x - 5)^2 + 4(5x - 5) - 4) + 3|) - 3) - 1 = (-3(2(-1x) + 5) - 3)/(1(2(-1x) + 5) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 196 | MATH |
[
{
"content": "How many solutions does the equation -3|(5(2|((-2x - 3)/(-3x + 2)) + 1| - 1)) - 2| - 3 = -3((1((-3x + 2)/(-1x - 2)) + 1)/(3((-3x + 2)/(-1x - 2)) - 1))^2 - 2((1((-3x + 2)/(-1x - 2)) + 1)/(3((-3x + 2)/(-1x - 2)) - 1)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2((-2(5(-1x^2 + 4x - 3) - 2) + 2)/(3(5(-1x^2 + 4x - 3) - 2) - 2)) + 2)/(-1((-2(5(-1x^2 + 4x - 3) - 2) + 2)/(3(5(-1x^2 + 4x - 3) - 2) - 2)) - 2) and g(x) = 2(-2sin(1π(-2x^2 - 5x - 2) - 2) - 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPre... | 404 | MATH |
[
{
"content": "How many solutions does the equation -1sin(1π(-2((-1(3x + 2) - 1)/(-2(3x + 2))) - 5) - 1) + 1 = (-1(-5(1x^2 - 2x - 5)^2 + 4(1x^2 - 2x - 5) + 2) + 3)/(2(-5(1x^2 - 2x - 5)^2 + 4(1x^2 - 2x - 5) + 2) - 1) have for x 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) = -3(1|(-4(2cos(2πx + 1) + 1) - 2) + 3| - 1)^2 - 5(1|(-4(2cos(2πx + 1) + 1) - 2) + 3| - 1) - 2 and g(x) = 3|((3(1|x + 1| - 3) - 1)/(-2(1|x + 1| - 3) - 3)) + 1| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\b... | 54 | MATH |
[
{
"content": "How many solutions does the equation -2sin(1π(1(-1(1x^2 - 1x + 3))^2 - 1(-1(1x^2 - 1x + 3)) + 3) + 3) + 3 = 3((2(-3sin(1πx + 1) - 1) - 3)/(2(-3sin(1πx + 1) - 1) + 2)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 441 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(3(-3|(-3sin(1πx + 2) + 3)| + 1)^2 - 5(-3|(-3sin(1πx + 2) + 3)| + 1) + 1) + 3| - 2 and g(x) = 2|(-1|(-2|x - 2| + 2) + 2| - 1) + 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 20 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(-3|(3|(5x + 4) - 1| - 2) + 1| - 3) - 1) + 2 = (1((2(3cos(3πx + 3) + 3) + 3)/(-2(3cos(3πx + 3) + 3) + 3)) - 2)/(-1((2(3cos(3πx + 3) + 3) + 3)/(-2(3cos(3πx + 3) + 3) + 3)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: ... | 423 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(2cos(1π(2|(-1|x + 3|)| + 1) + 3) + 2) + 5 and g(x) = -2cos(3π(-1|(-1x - 1) - 3| + 1) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(1(4(2|x + 3| - 2) + 2)^2 + 2(4(2|x + 3| - 2) + 2) - 5))/(-1(1(4(2|x + 3| - 2) + 2)^2 + 2(4(2|x + 3| - 2) + 2) - 5) - 1) and g(x) = -2e^(-1(2(2|x - 3| - 1) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((-1(-2|((-1x - 2)/(1x)) + 1|) - 3)/(3(-2|((-1x - 2)/(1x)) + 1|) + 1))^2 + 5((-1(-2|((-1x - 2)/(1x)) + 1|) - 3)/(3(-2|((-1x - 2)/(1x)) + 1|) + 1)) and g(x) = -3((3(-1cos(2πx - 3) - 2) + 1)/(1(-1cos(2πx - 3) - 2) + 3)) intersect in the interval [-10, 1... | 30 | MATH |
[
{
"content": "How many solutions does the equation (3(-5(-2e^(2(-1|x| - 2)) + 3) + 1) - 3)/(2(-5(-2e^(2(-1|x| - 2)) + 3) + 1) + 1) = -1cos(3π(-2(-5x^2 - 5x + 1)^2 - 4(-5x^2 - 5x + 1) - 5) + 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 802 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-3cos(3π(1((-3x + 2)/(-1x + 2))^2 + 3((-3x + 2)/(-1x + 2)) + 2)) + 1) + 2 and g(x) = 4(1sin(1π(-4x - 5) - 3) - 3) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 170 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-1|((1(1|x + 1| - 3) + 1)/(3(1|x + 1| - 3) + 1))| + 1) + 1)/(1(-1|((1(1|x + 1| - 3) + 1)/(3(1|x + 1| - 3) + 1))| + 1) + 3) and g(x) = (-2(-4(-2e^(1x) + 2)^2 - 5(-2e^(1x) + 2) + 4) - 2)/(-2(-4(-2e^(1x) + 2)^2 - 5(-2e^(1x) + 2) + 4) + 1) intersect in... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2((3(3(-1e^(1x) - 1) + 4) + 3)/(-2(3(-1e^(1x) - 1) + 4))) + 5 and g(x) = -5(2cos(2π(1|x - 2| + 3) - 1) - 3)^2 - 2(2cos(2π(1|x - 2| + 3) - 1) - 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-2|(3((-1x - 3)/(2x)) - 1) - 3| + 3) - 3)/(-3(-2|(3((-1x - 3)/(2x)) - 1) - 3| + 3) + 4) and g(x) = -1|(2|(2sin(3πx + 1) - 3)| + 1) - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 59 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-3(1(4x^2 - 2x + 5)^2 - 5(4x^2 - 2x + 5) - 1)^2 + 4) - 5 and g(x) = (3((1(-3|x - 1| - 2) + 3)/(-1(-3|x - 1| - 2) + 2)) + 1)/(1((1(-3|x - 1| - 2) + 3)/(-1(-3|x - 1| - 2) + 2)) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer ... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1(-3sin(3π((-2(1|x - 3| - 1) - 3)/(2(1|x - 3| - 1) - 3))) - 2) - 4 = (3(3|(3sin(2πx - 3) + 2) + 1| - 1) + 3)/(-3(3|(3sin(2πx - 3) + 2) + 1| - 1) + 4) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 212 | MATH |
[
{
"content": "How many solutions does the equation 1|(2e^(1((1(1x^2 - 1x + 5) + 2)/(-3(1x^2 - 1x + 5) + 4))) - 3) - 2| - 2 = 3cos(3π(-5(1x - 4)^2 - 3(1x - 4) - 4) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1445 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(5((-3(2|x + 1|))/(2(2|x + 1|) + 3)) - 5)^2 + 1(5((-3(2|x + 1|))/(2(2|x + 1|) + 3)) - 5) - 5 and g(x) = 3|(1|(-1x^2 + 2x - 2) - 3| + 3) - 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-1(-1|(-4(-2|x - 1| - 2)^2 - 4(-2|x - 1| - 2) + 3) + 3| + 3) + 3)/(3(-1|(-4(-2|x - 1| - 2)^2 - 4(-2|x - 1| - 2) + 3) + 3| + 3) - 2) = 1(3cos(3π(-2x^2 - 5x + 4) + 1) + 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Y... | 1077 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-2(2(3e^(1x) - 3) - 5) - 1) - 3 and g(x) = (-3(-4(-2cos(2πx - 1) - 2)^2 + 1(-2cos(2πx - 1) - 2) + 4) + 1)/(-1(-4(-2cos(2πx - 1) - 2)^2 + 1(-2cos(2πx - 1) - 2) + 4) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex for... | 23 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(2π(2((1((2x + 1)/(-1x - 2)) - 1)/(-2((2x + 1)/(-1x - 2)) - 3))^2 - 4((1((2x + 1)/(-1x - 2)) - 1)/(-2((2x + 1)/(-1x - 2)) - 3)) + 3)) - 3 and g(x) = -2((-3(-3sin(3πx - 1) + 1) + 2)/(-2(-3sin(3πx - 1) + 1) - 2)) - 4 intersect in the interval [-10, ... | 78 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3((-2(4((3x + 3)/(2x)) + 3) - 2)/(-2(4((3x + 3)/(2x)) + 3) + 1)) + 1)/(-3((-2(4((3x + 3)/(2x)) + 3) - 2)/(-2(4((3x + 3)/(2x)) + 3) + 1)) + 2) and g(x) = 2(3sin(3π(4x - 4) - 3))^2 - 5(3sin(3π(4x - 4) - 3)) - 4 intersect in the interval [-10, 10].\n\nPr... | 472 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(1|(-1(-3sin(1πx + 3) + 1)^2 + 2(-3sin(1πx + 3) + 1) - 3) - 1| - 2)^2 - 2(1|(-1(-3sin(1πx + 3) + 1)^2 + 2(-3sin(1πx + 3) + 1) - 3) - 1| - 2) - 2 and g(x) = 1|((-3(1|x + 1| - 3) - 2)/(3(1|x + 1| - 3) + 2)) - 3| + 3 in the range -10... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-4((3(-3sin(2πx + 1) - 1) - 2)/(-1(-3sin(2πx + 1) - 1) + 2)) - 4)^2 + 2(-4((3(-3sin(2πx + 1) - 1) - 2)/(-1(-3sin(2πx + 1) - 1) + 2)) - 4) - 1 and g(x) = 2sin(1π(3(3x)^2 - 2(3x) - 4) - 2) + 1 where -10 ≤ x ≤ 10.\n\nPresent t... | 89 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(2sin(2π((1((2x + 1)/(2x - 1)) - 2)/(-3((2x + 1)/(2x - 1)) + 2)) + 2) - 1) + 1)/(-1(2sin(2π((1((2x + 1)/(2x - 1)) - 2)/(-3((2x + 1)/(2x - 1)) + 2)) + 2) - 1) + 2) and g(x) = -1(3|(-1x - 3) + 2| - 2) + 5 in the range -10 ≤ x ≤ 10?... | 44 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(2(1cos(1π((-1x + 1)/(1x)) + 3) - 3)^2 - 3(1cos(1π((-1x + 1)/(1x)) + 3) - 3) - 3) + 3| - 2 and g(x) = -2|((3(3x^2 - 4x) - 2)/(-2(3x^2 - 4x) - 1)) - 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(2(-3(-3x^2 - 2x + 3) - 5))^2 - 4(2(-3(-3x^2 - 2x + 3) - 5)) - 2 and g(x) = -4(1(-1|x| + 1) + 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) = -2((-1(-3(-2x)^2 + 5(-2x) - 4))/(-3(-3(-2x)^2 + 5(-2x) - 4) - 2))^2 + 5((-1(-3(-2x)^2 + 5(-2x) - 4))/(-3(-3(-2x)^2 + 5(-2x) - 4) - 2)) + 5 and g(x) = -1(-4(2cos(3πx - 2))^2 + 3(2cos(3πx - 2))) - 2 where -10 ≤ x ≤ 10.\n\nPrese... | 120 | MATH |
[
{
"content": "How many solutions does the equation -1((1((3(3sin(3πx) + 3) + 1)/(2(3sin(3πx) + 3) - 3)))/(1((3(3sin(3πx) + 3) + 1)/(2(3sin(3πx) + 3) - 3)) + 2))^2 - 4((1((3(3sin(3πx) + 3) + 1)/(2(3sin(3πx) + 3) - 3)))/(1((3(3sin(3πx) + 3) + 1)/(2(3sin(3πx) + 3) - 3)) + 2)) + 2 = (2((1(4x^2 - 2x + 1) - 2)/(1(4x^... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(1e^(-1(-1(4x^2 - 3x - 2) - 1))) - 3| - 3 and g(x) = 3cos(3π(3(1x - 3)) - 3) + 3 in the range -10 ≤ x ≤ 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) = 5(-2|(-3|(-1|x - 1| + 3) + 3| + 2) - 2| - 1) - 1 and g(x) = (1(1(3cos(3πx + 2) + 2)^2 + 5(3cos(3πx + 2) + 2) + 2) - 3)/(-1(1(3cos(3πx + 2) + 2)^2 + 5(3cos(3πx + 2) + 2) + 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(1(2|(-3x - 2)| - 1)^2 - 1(2|(-3x - 2)| - 1) - 1) + 1| + 1 and g(x) = -2sin(2π((1(2|x - 3| - 1) - 3)/(-2(2|x - 3| - 1) + 1)) - 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-2sin(2π(1|((1x)/(3x - 2)) - 3|) + 3) + 3)^2 - 5(-2sin(2π(1|((1x)/(3x - 2)) - 3|) + 3) + 3) + 1 and g(x) = -2(-3sin(1π((2x - 2)/(2x + 3)) - 2) - 2)^2 + 5(-3sin(1π((2x - 2)/(2x + 3)) - 2) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the a... | 59 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-2|((2((1x + 2)/(-1x - 1)) + 1)/(3((1x + 2)/(-1x - 1)) - 1)) - 1|) + 4 and g(x) = -5(1|((3x)/(1x - 2)) - 3| - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-3e^(-2(5(-3x^2 + 5x - 1)^2 + 4(-3x^2 + 5x - 1))) - 1) + 2)/(2(-3e^(-2(5(-3x^2 + 5x - 1)^2 + 4(-3x^2 + 5x - 1))) - 1) + 2) and g(x) = (1((2(-1sin(3πx) - 3) - 1)/(2(-1sin(3πx) - 3) + 3)) + 1)/(-2((2(-1sin(3πx) - 3) - 1)/(2(-1sin(3πx) - 3) + 3)) + 3)... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|((-2(1|(-1cos(1πx + 2) - 1) + 1| - 1) + 2)/(1(1|(-1cos(1πx + 2) - 1) + 1| - 1) - 2)) - 1| - 3 and g(x) = 2((3(-3sin(1πx)) + 2)/(1(-3sin(1πx)) - 3)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(1π((3((-2((-3x + 2)/(2x + 1)) - 2)/(2((-3x + 2)/(2x + 1)))) + 1)/(-2((-2((-3x + 2)/(2x + 1)) - 2)/(2((-3x + 2)/(2x + 1)))) + 1)) - 1) and g(x) = 1cos(2π(3|(5x + 3)| + 3) + 2) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer... | 352 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-1cos(1π(1(-4x^2 - 3x - 3) + 4) - 1) - 2) + 1| - 1 and g(x) = 1|((-1(3sin(1πx + 2) + 2) + 3)/(-3(3sin(1πx + 2) + 2) + 1)) - 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 711 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(2sin(3π(-1|(-1|x + 3| + 1) + 3| - 2) + 1) - 2) + 3)/(-1(2sin(3π(-1|(-1|x + 3| + 1) + 3| - 2) + 1) - 2) + 3) and g(x) = 4(-3cos(1π(-4x + 5) + 2) - 3)^2 - 5(-3cos(1π(-4x + 5) + 2) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the an... | 80 | MATH |
[
{
"content": "How many solutions does the equation (2(-1(1(5x^2 + 3x - 4) - 1) + 5) + 1)/(-3(-1(1(5x^2 + 3x - 4) - 1) + 5) - 3) = -2|(2((-1x - 1)/(-3x + 1))^2 - 3((-1x - 1)/(-3x + 1)) + 1) + 1| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-3((2(2x + 2) + 2)/(-3(2x + 2) + 1)) - 5) - 1| + 3 and g(x) = -2(1((2x - 1)/(-1x + 3))^2 + 4((2x - 1)/(-1x + 3)) - 1) - 4 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) = 1((-2((-1(-1x^2 + 4x - 2) - 3)/(-3(-1x^2 + 4x - 2) - 2)))/(-1((-1(-1x^2 + 4x - 2) - 3)/(-3(-1x^2 + 4x - 2) - 2)) + 3))^2 - 3((-2((-1(-1x^2 + 4x - 2) - 3)/(-3(-1x^2 + 4x - 2) - 2)))/(-1((-1(-1x^2 + 4x - 2) - 3)/(-3(-1x^2 + 4x ... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-5(-1(-1x^2 - 2x + 2)^2 - 4(-1x^2 - 2x + 2) + 1)) - 5 and g(x) = 3(2((2x - 3)/(-2x - 2)) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(3((2(-4x + 5))/(-3(-4x + 5) - 3))^2 - 5((2(-4x + 5))/(-3(-4x + 5) - 3)) - 1) - 2)/(-1(3((2(-4x + 5))/(-3(-4x + 5) - 3))^2 - 5((2(-4x + 5))/(-3(-4x + 5) - 3)) - 1) + 2) and g(x) = 2(2cos(1π(-5x - 5) + 3) + 2) + 5 where -10... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-2(3sin(3π(-3|x + 3| + 2)) + 2)^2 - 1(3sin(3π(-3|x + 3| + 2)) + 2) - 4) + 2 and g(x) = (-2(3(2|x + 3| - 2)^2 - 4(2|x + 3| - 2) - 2) - 3)/(1(3(2|x + 3| - 2)^2 - 4(2|x + 3| - 2) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-1e^(2((-1(1x^2 - 2x))/(1(1x^2 - 2x) + 1))) + 2) + 2)/(1(-1e^(2((-1(1x^2 - 2x))/(1(1x^2 - 2x) + 1))) + 2) + 3) and g(x) = 3|((-1(1x^2 - 3x - 5) + 2)/(2(1x^2 - 3x - 5) - 1)) - 1| where -10 ≤ x ≤ 10.\n\nPresent the answer in... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(1π(2(-4(3|x - 3| + 3) + 3)^2 - 3(-4(3|x - 3| + 3) + 3) + 4) - 1) - 1 and g(x) = 5(-2e^(1(5x - 5)))^2 + 4(-2e^(1(5x - 5))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 873 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-3|(-2|(2x - 3) + 1|) + 3| + 1)^2 - 2(-3|(-2|(2x - 3) + 1|) + 3| + 1) + 1 and g(x) = 1(3(-1cos(3πx) + 1)^2 - 3(-1cos(3πx) + 1) - 2) - 2 in the range -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) = -1|((2(-2((1x - 3)/(-3x - 2))^2 + 1) - 1)/(3(-2((1x - 3)/(-3x - 2))^2 + 1) - 2)) - 3| - 3 and g(x) = -1|(-4(-1x - 3)^2 - 5(-1x - 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) = -5(-1|((-3(1x^2 - 4))/(-3(1x^2 - 4) - 1)) - 3| + 2) - 2 and g(x) = -4(-2(1x + 2) - 4) - 5 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) = -1(-4((-3(-3x^2 + 2x + 4) + 2)/(-2(-3x^2 + 2x + 4))))^2 - 5(-4((-3(-3x^2 + 2x + 4) + 2)/(-2(-3x^2 + 2x + 4)))) + 4 and g(x) = -3|((-2(-5x^2 - 2x + 4) - 1)/(1(-5x^2 - 2x + 4) - 3)) + 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in L... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(3((-1(3x - 5) + 1)/(-1(3x - 5) - 1))^2 - 4((-1(3x - 5) + 1)/(-1(3x - 5) - 1)) + 2) - 3 and g(x) = 3|((2(-4x - 4) - 1)/(2(-4x - 4) + 1)) + 2| - 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) = -2(-3cos(3π(1|(3|x + 2|) + 2| - 1) - 3))^2 + 3(-3cos(3π(1|(3|x + 2|) + 2| - 1) - 3)) and g(x) = -1cos(1π(2|(5x^2 - 2x) - 3| + 1) - 1) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 371 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|((3(1(-1x^2) - 2))/(1(1(-1x^2) - 2) + 3)) + 3| + 1 and g(x) = -2|(-5(-4x^2 - 2x - 5) + 1) + 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 (2(1(-3sin(3π(5x^2 - 2x + 1) + 3) - 3)^2 + 4(-3sin(3π(5x^2 - 2x + 1) + 3) - 3) + 2) - 1)/(-2(1(-3sin(3π(5x^2 - 2x + 1) + 3) - 3)^2 + 4(-3sin(3π(5x^2 - 2x + 1) + 3) - 3) + 2) - 2) = 2((-3((-3x)/(1x + 2)) + 2)/(2((-3x)/(1x + 2)) - 1)) have for x in the interval [... | 469 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((1(4(-5x^2 + 1x + 3)^2 - 3(-5x^2 + 1x + 3) - 4) - 1)/(-2(4(-5x^2 + 1x + 3)^2 - 3(-5x^2 + 1x + 3) - 4)))^2 + 4((1(4(-5x^2 + 1x + 3)^2 - 3(-5x^2 + 1x + 3) - 4) - 1)/(-2(4(-5x^2 + 1x + 3)^2 - 3(-5x^2 + 1x + 3) - 4))) + 5 and ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(2((-1(1|x - 2|) - 2)/(3(1|x - 2|) - 2))^2 + 2((-1(1|x - 2|) - 2)/(3(1|x - 2|) - 2)) - 1) + 1 and g(x) = 5((-3(-2sin(2πx + 3) + 1) - 3)/(3(-2sin(2πx + 3) + 1) + 2))^2 + 5((-3(-2sin(2πx + 3) + 1) - 3)/(3(-2sin(2πx + 3) + 1) + 2)) +... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(2(4(1cos(1πx + 3) + 2)^2 + 4(1cos(1πx + 3) + 2) + 1) - 3) - 2)/(3(2(4(1cos(1πx + 3) + 2)^2 + 4(1cos(1πx + 3) + 2) + 1) - 3) + 3) and g(x) = -2|(1|((1x + 2)/(3x)) + 3| - 1) - 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-2|(1cos(3π((-3x - 1)/(-2x - 3)) - 1) - 2) + 3| + 1) - 3| + 3 and g(x) = (1(2|(2sin(1πx - 3) - 1) - 1| + 2) - 1)/(1(2|(2sin(1πx - 3) - 1) - 1| + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(5(-2|(-3|x| + 1) + 2| + 3) - 2) - 4 and g(x) = 2|(2|(1|x + 2|) + 3|) + 3| + 3 where -10 ≤ x ≤ 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) = 3e^(-1(3|(2((1x + 3)/(2x)) - 5) + 1|)) - 3 and g(x) = 5(-5(3x^2 - 5x - 5)^2 - 4(3x^2 - 5x - 5) + 3) - 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 -2|(3e^(2(4(-4x^2 - 4x + 4)^2 - 5(-4x^2 - 4x + 4) + 1)) + 1) - 2| + 2 = 2e^(1(-2|(-3x - 2)| + 2)) - 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 solutions does the equation (2((2(3|(-2|x| + 2) - 2| + 1) - 2)/(-2(3|(-2|x| + 2) - 2| + 1))) - 1)/(-1((2(3|(-2|x| + 2) - 2| + 1) - 2)/(-2(3|(-2|x| + 2) - 2| + 1))) + 2) = -4(1(-1x^2 + 5x - 4)^2 + 5(-1x^2 + 5x - 4) - 5) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaT... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(1((-2(-2cos(1πx) + 3) + 2)/(-1(-2cos(1πx) + 3) + 2))^2 + 5((-2(-2cos(1πx) + 3) + 2)/(-1(-2cos(1πx) + 3) + 2)) + 4)^2 - 4(1((-2(-2cos(1πx) + 3) + 2)/(-1(-2cos(1πx) + 3) + 2))^2 + 5((-2(-2cos(1πx) + 3) + 2)/(-1(-2cos(1πx) + 3) + 2)... | 13 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-1|((-3(-1x) - 3)/(-1(-1x) - 2))| + 3) + 1)/(2(-1|((-3(-1x) - 3)/(-1(-1x) - 2))| + 3)) and g(x) = 5(-3sin(1π(-3|x - 3| - 2) - 2) - 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 61 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((-1(-3(3x^2 + 1x - 2)^2 - 1(3x^2 + 1x - 2) + 4) + 3)/(3(-3(3x^2 + 1x - 2)^2 - 1(3x^2 + 1x - 2) + 4) + 1)) + 3 and g(x) = 2cos(2π(3(-3x + 4)^2 - 1(-3x + 4) + 5) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\box... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((1(3cos(2π(2x - 3) - 3) - 3) + 1)/(-3(3cos(2π(2x - 3) - 3) - 3) - 2)) + 3 and g(x) = 1|(-3(2x + 5) + 5) + 2| 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 (-2((-3(3(-3x + 5)^2 + 2(-3x + 5) - 5) - 2)/(-3(3(-3x + 5)^2 + 2(-3x + 5) - 5) + 4)))/(3((-3(3(-3x + 5)^2 + 2(-3x + 5) - 5) - 2)/(-3(3(-3x + 5)^2 + 2(-3x + 5) - 5) + 4)) + 2) = (2(-2sin(3π(-2x - 3) + 2) + 1))/(3(-2sin(3π(-2x - 3) + 2) + 1) + 2) have for x in th... | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(1π((3(1(-3|x + 3| + 1) + 5) - 3)/(-1(1(-3|x + 3| + 1) + 5) - 1)) - 3) - 3 and g(x) = 2cos(3π(4((3x)/(-1x + 2)) - 1)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 280 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(1|(3(-2sin(2πx + 2) + 2) - 4) - 3| + 3) + 3 and g(x) = 1(-1((2x)/(-2x - 3)) + 3)^2 + 3(-1((2x)/(-2x - 3)) + 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 66 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(3|(-3cos(1π(-3|x + 3| + 3) - 1))| - 2) - 3)/(2(3|(-3cos(1π(-3|x + 3| + 3) - 1))| - 2) + 3) and g(x) = -2(4(1sin(2πx - 1)) + 5)^2 + 4(4(1sin(2πx - 1)) + 5) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your a... | 48 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-1sin(2π((2(3|x + 1|) + 1)/(1(3|x + 1|) + 1)) - 3) + 3) - 5 and g(x) = (-2(3cos(1π(4x^2 + 2x + 5)) + 2) + 2)/(-3(3cos(1π(4x^2 + 2x + 5)) + 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 134 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(4(1(-2x - 1) + 2)^2 + 2(1(-2x - 1) + 2) - 3)^2 + 5(4(1(-2x - 1) + 2)^2 + 2(1(-2x - 1) + 2) - 3) - 5 and g(x) = 2cos(3π(-4((1x - 1)/(3x)) + 1) + 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(1((-2(-2sin(2πx - 1)) - 3)/(-1(-2sin(2πx - 1)) - 2))^2 + 5((-2(-2sin(2πx - 1)) - 3)/(-1(-2sin(2πx - 1)) - 2)) - 5) - 1 and g(x) = 3|((1((-1x + 1)/(1x - 2)))/(2((-1x + 1)/(1x - 2)) + 1)) + 1| in the range -10 ≤ x ≤ 10?\n\nPresent ... | 78 | MATH |
[
{
"content": "How many solutions does the equation -4(-3(-1(-1|x + 1| - 1) - 2) - 2)^2 - 3(-3(-1(-1|x + 1| - 1) - 2) - 2) - 1 = -1cos(3π(1|((-1x - 2)/(2x - 1)) - 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": "Determine the number of intersection points between the functions f(x) = (2((3(1((-1x - 3)/(-1x - 3)) + 3) + 1)/(-2(1((-1x - 3)/(-1x - 3)) + 3) + 1)) - 1)/(-2((3(1((-1x - 3)/(-1x - 3)) + 3) + 1)/(-2(1((-1x - 3)/(-1x - 3)) + 3) + 1)) - 1) and g(x) = 1(3e^(-2(-1|x - 1| + 1)) - 3)^2 - 5(3e^(-2(-1|x -... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(3π(-3(3((-2x + 1)/(2x)) - 3) + 4) - 2) - 2 and g(x) = -2(-3|(1|x + 1|)| + 1)^2 - 5(-3|(1|x + 1|)| + 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "How many solutions does the equation -4(1(2cos(1π((3x - 2)/(3x + 1)) - 3))) = 1(-2|(-1cos(1πx + 2) - 1) - 2|)^2 + 5(-2|(-1cos(1πx + 2) - 1) - 2|) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(2|((-2(1x^2 - 2x - 5) + 1)/(-2(1x^2 - 2x - 5) + 3)) - 2| - 3)^2 - 4(2|((-2(1x^2 - 2x - 5) + 1)/(-2(1x^2 - 2x - 5) + 3)) - 2| - 3) + 4 and g(x) = -4((1(5x - 4))/(3(5x - 4) + 2))^2 + 1((1(5x - 4))/(3(5x - 4) + 2)) + 1 intersect in the interval [-10, 10... | 8 | MATH |
[
{
"content": "How many solutions does the equation (1(-2|(2(3|x + 1|) + 5) + 1| - 2) - 2)/(-3(-2|(2(3|x + 1|) + 5) + 1| - 2) + 2) = -4(-4(-2|x - 2|)^2 + 2(-2|x - 2|) + 5)^2 - 2(-4(-2|x - 2|)^2 + 2(-2|x - 2|) + 5) + 2 have for x in the interval [-10, 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) = -1(3(-1(-3|x + 3|)^2 + 4(-3|x + 3|) + 5)^2 + 3(-1(-3|x + 3|)^2 + 4(-3|x + 3|) + 5) - 3) + 2 and g(x) = (2((-3(5x^2 - 3x + 4) + 3)/(2(5x^2 - 3x + 4) - 1)) + 1)/(2((-3(5x^2 - 3x + 4) + 3)/(2(5x^2 - 3x + 4) - 1)) + 1) in the range -10... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-3(1|(5x - 1) + 1| - 1) - 5) + 1| + 3 and g(x) = 1cos(1π(3((-1x - 2)/(2x - 1))^2 + 5((-1x - 2)/(2x - 1)) + 3) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-3|(2(-5x + 4) + 4) + 1| + 2) - 2)/(-2(-3|(2(-5x + 4) + 4) + 1| + 2) + 3) and g(x) = (1(-2cos(3π(-3x^2 + 3x - 2) - 1) + 1) - 1)/(-1(-2cos(3π(-3x^2 + 3x - 2) - 1) + 1) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTe... | 1427 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(1sin(2π(-1(1x^2 + 1x - 5)^2 + 2(1x^2 + 1x - 5) - 3) + 1) - 1) - 2)/(-3(1sin(2π(-1(1x^2 + 1x - 5)^2 + 2(1x^2 + 1x - 5) - 3) + 1) - 1) + 4) and g(x) = 2sin(1π(-5(-2|x - 3| - 3)^2 - 4(-2|x - 3| - 3) + 1) + 1) in the range -10 ≤ x ... | 1528 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-3sin(2π((-1((-1x - 3)/(2x - 1)) + 1)/(2((-1x - 3)/(2x - 1)) + 1)) - 2)) + 2)/(2(-3sin(2π((-1((-1x - 3)/(2x - 1)) + 1)/(2((-1x - 3)/(2x - 1)) + 1)) - 2)) + 2) and g(x) = -1((1(3e^(1x) - 2) + 2)/(2(3e^(1x) - 2) + 1))^2 + 5((1(3e^... | 19 | MATH |
[
{
"content": "How many solutions does the equation -2|(-3|((-1(-1|x + 3|) - 2)/(3(-1|x + 3|) + 3)) - 1| + 3) - 2| = -3(-1(3sin(3πx - 2) - 3)^2 + 2)^2 + 1(-1(3sin(3πx - 2) - 3)^2 + 2) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 72 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-5(2(3|x + 1| - 2)^2 + 5)^2 + 4(2(3|x + 1| - 2)^2 + 5) - 5))/(3(-5(2(3|x + 1| - 2)^2 + 5)^2 + 4(2(3|x + 1| - 2)^2 + 5) - 5) + 2) and g(x) = -3cos(2π(3|(-1x - 1) + 2| + 3) - 2) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answe... | 119 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(1π(-5(5(5x) - 3) + 3) - 2) - 1 and g(x) = -2(-1e^(2(-4x - 4)))^2 + 3(-1e^(2(-4x - 4))) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1099 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(3π(-2|(-3((-2x - 1)/(1x)) + 3) - 3| + 1) - 3) - 3 and g(x) = 2sin(2π(-5(1|x| + 3)^2 - 4(1|x| + 3) - 3) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 852 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(2π(-3(-2|(1x + 2) + 2| + 2) + 4) - 3) - 2 and g(x) = 1e^(-1(1(4x) - 5)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 107 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-1(3cos(3π(2x^2 - 4x + 3) + 1) - 3)^2 + 2) + 4 and g(x) = 2(2|(-2sin(2πx + 3) + 1) - 3| - 1) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 298 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-2(-1(-5x^2 + 4x + 2)^2 - 5(-5x^2 + 4x + 2) + 2)^2 - 2(-1(-5x^2 + 4x + 2)^2 - 5(-5x^2 + 4x + 2) + 2) + 4)^2 - 2(-2(-1(-5x^2 + 4x + 2)^2 - 5(-5x^2 + 4x + 2) + 2)^2 - 2(-1(-5x^2 + 4x + 2)^2 - 5(-5x^2 + 4x + 2) + 2) + 4) and g(x) = 2|(-2sin(1π((3x + 1)/... | 16 | MATH |
[
{
"content": "How many solutions does the equation 5(-2(1(4x - 2)^2 + 1(4x - 2) + 1)^2 + 4(1(4x - 2)^2 + 1(4x - 2) + 1) - 4) + 3 = -4(-1sin(3π(3x - 5) - 2) + 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(2π(2(3((2x - 2)/(-1x + 3))^2 + 1((2x - 2)/(-1x + 3)) - 5)^2 + 5(3((2x - 2)/(-1x + 3))^2 + 1((2x - 2)/(-1x + 3)) - 5) - 4) - 1) - 1 and g(x) = (2(-2|(2cos(2πx + 1) + 2) + 1| + 3) + 2)/(1(-2|(2cos(2πx + 1) + 2) + 1| + 3) - 3) in... | 382 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(3(2sin(2π((3x - 3)/(2x)) - 1) - 3)^2 + 4(2sin(2π((3x - 3)/(2x)) - 1) - 3) + 4) - 1 and g(x) = (3((3(-3sin(1πx + 1) + 1) - 3)/(2(-3sin(1πx + 1) + 1) + 3)) + 1)/(2((3(-3sin(1πx + 1) + 1) - 3)/(2(-3sin(1πx + 1) + 1) + 3)) - 1) intersect in the interval ... | 20 | MATH |
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