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) = (1(2|(1(1x + 4)^2 - 3(1x + 4) - 4) - 3| + 2) + 3)/(-1(2|(1(1x + 4)^2 - 3(1x + 4) - 4) - 3| + 2)) and g(x) = -2|(-4(-5x^2 - 4x + 1)^2 + 5(-5x^2 - 4x + 1) + 2) + 1| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format:... | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-5(5(4x^2 + 3x - 4) - 2)^2 - 3(5(4x^2 + 3x - 4) - 2) + 1))/(-1(-5(5(4x^2 + 3x - 4) - 2)^2 - 3(5(4x^2 + 3x - 4) - 2) + 1) + 2) and g(x) = (2((-1(-3|x - 3| - 3) + 1)/(1(-3|x - 3| - 3) - 2)) + 2)/(-3((-1(-3|x - 3| - 3) + 1)/... | 4 | MATH |
[
{
"content": "How many solutions does the equation 5(1e^(1(-4(-2|x + 3|)^2 - 4(-2|x + 3|) - 1)) + 2)^2 - 4(1e^(1(-4(-2|x + 3|)^2 - 4(-2|x + 3|) - 1)) + 2) - 4 = (-2(-1(-1sin(2πx - 2) + 2) + 2) + 2)/(3(-1(-1sin(2πx - 2) + 2) + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3|((-1(1e^(1x) + 3) + 1)/(1(1e^(1x) + 3) - 3))| + 3) + 5 and g(x) = 5(2(1x^2 - 3x + 1)^2 + 1(1x^2 - 3x + 1))^2 - 1(2(1x^2 - 3x + 1)^2 + 1(1x^2 - 3x + 1)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your an... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1((-2((-1(-1x - 3) + 2)/(1(-1x - 3) - 2)) - 3)/(2((-1(-1x - 3) + 2)/(1(-1x - 3) - 2)) + 3)) + 3)/(3((-2((-1(-1x - 3) + 2)/(1(-1x - 3) - 2)) - 3)/(2((-1(-1x - 3) + 2)/(1(-1x - 3) - 2)) + 3)) - 2) and g(x) = -2(-3(4x^2 - 2x - 2)^2 + 2(4x^2 - 2x - 2) + ... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-3(-3sin(2π(-1|x| - 1) - 2) + 2)^2 + 5(-3sin(2π(-1|x| - 1) - 2) + 2) + 2)^2 + 5(-3(-3sin(2π(-1|x| - 1) - 2) + 2)^2 + 5(-3sin(2π(-1|x| - 1) - 2) + 2) + 2) + 4 and g(x) = 3(-1(4x - 2) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer... | 46 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((-3(-2sin(1π(-2x + 3) + 1) + 1) - 2)/(-1(-2sin(1π(-2x + 3) + 1) + 1) + 3))^2 + 1((-3(-2sin(1π(-2x + 3) + 1) + 1) - 2)/(-1(-2sin(1π(-2x + 3) + 1) + 1) + 3)) + 4 and g(x) = 1|(-3|((-3x - 1)/(1x + 3)) + 2|)| - 1 where -10 ≤ x... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(2|((-1(-2x - 2) - 2)/(3(-2x - 2) + 1)) + 3| - 3)^2 + 3(2|((-1(-2x - 2) - 2)/(3(-2x - 2) + 1)) + 3| - 3) and g(x) = (-2((-1(-5x^2 - 3x + 4) + 3)/(3(-5x^2 - 3x + 4))) + 3)/(1((-1(-5x^2 - 3x + 4) + 3)/(3(-5x^2 - 3x + 4))) - 3) in t... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(3π(-3|(3((-3x + 3)/(2x - 3)) - 4) - 2| + 2) + 2) + 2 and g(x) = (2(-1e^(1(-2x^2 - 5x - 3)) + 1) - 2)/(-3(-1e^(1(-2x^2 - 5x - 3)) + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 68 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π(-1(3(1|x - 3| - 3) - 5)^2 - 1(3(1|x - 3| - 3) - 5) + 1) + 1) + 2 = -1(3(-3e^(-1x) + 3) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(3|(5((-3x + 3)/(-2x + 1))) - 1| - 3)^2 - 3(3|(5((-3x + 3)/(-2x + 1))) - 1| - 3) + 2 and g(x) = 1|((-2(1x^2 + 4x + 3) + 2)/(1(1x^2 + 4x + 3) + 2)) - 3| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2(3(-2|(3|x|) - 3| + 2) + 1) + 3)/(-2(3(-2|(3|x|) - 3| + 2) + 1) - 1) = 2(1(-2cos(1πx) - 1)^2 - 3(-2cos(1πx) - 1) + 2) + 5 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 -2sin(1π(-1(-4(-3x - 2) - 3) - 1) - 1) - 2 = (-3(1(-5x^2 - 5x + 3)^2 + 4(-5x^2 - 5x + 3) + 1))/(-2(1(-5x^2 - 5x + 3)^2 + 4(-5x^2 - 5x + 3) + 1) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(2cos(1π(-2(-2x)^2 - 4(-2x) - 2) + 1) - 1)| - 3 and g(x) = -4((-2(-1x^2 + 5x - 3) - 2)/(-2(-1x^2 + 5x - 3) + 3))^2 - 5((-2(-1x^2 + 5x - 3) - 2)/(-2(-1x^2 + 5x - 3) + 3)) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in La... | 980 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2((3(3e^(-2(-4x^2 - 1x + 5)) - 1) - 3)/(3(3e^(-2(-4x^2 - 1x + 5)) - 1) + 1))^2 + 3((3(3e^(-2(-4x^2 - 1x + 5)) - 1) - 3)/(3(3e^(-2(-4x^2 - 1x + 5)) - 1) + 1)) - 2 and g(x) = 3(3|((2x + 3)/(2x))| + 1)^2 - 3(3|((2x + 3)/(2x))| + 1) - ... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(-4((1(-1|x + 1| + 2) - 1)/(-2(-1|x + 1| + 2) - 2))^2 + 4((1(-1|x + 1| + 2) - 1)/(-2(-1|x + 1| + 2) - 2)) + 4)| - 3 and g(x) = -3(-1e^(2(-2|x - 3| - 2))) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxe... | 6 | MATH |
[
{
"content": "How many solutions does the equation 3|(3(-1e^(-2(2x^2 - 5x - 1)) - 1) - 2)| + 1 = 2|(2(2|x + 1| - 3)^2 - 5(2|x + 1| - 3) - 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 solutions does the equation (2(1(-2cos(1π(4x - 2) - 1) + 2)^2 + 4(-2cos(1π(4x - 2) - 1) + 2) - 3) + 3)/(-3(1(-2cos(1π(4x - 2) - 1) + 2)^2 + 4(-2cos(1π(4x - 2) - 1) + 2) - 3)) = -2(-1(-3|x + 1| + 1)^2 - 4(-3|x + 1| + 1) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer i... | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(1(2cos(2π(-4x + 3) + 3) + 3)^2 - 4) - 1 and g(x) = -2sin(2π(5(-2x^2 + 2x + 3) - 4) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 168 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(2(1(-1x^2 - 2x + 2)^2 + 3(-1x^2 - 2x + 2) - 5) + 3) - 1)/(-2(2(1(-1x^2 - 2x + 2)^2 + 3(-1x^2 - 2x + 2) - 5) + 3) + 1) and g(x) = 1(-4(-1sin(1πx + 2) + 1) - 2) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format:... | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3((-1((2(-2e^(-1x) - 1) - 1)/(2(-2e^(-1x) - 1))) + 2)/(3((2(-2e^(-1x) - 1) - 1)/(2(-2e^(-1x) - 1))))))/(2((-1((2(-2e^(-1x) - 1) - 1)/(2(-2e^(-1x) - 1))) + 2)/(3((2(-2e^(-1x) - 1) - 1)/(2(-2e^(-1x) - 1))))) - 3) and g(x) = -3sin(2π... | 80 | MATH |
[
{
"content": "How many solutions does the equation 2|((-3((-2(-4x) - 2)/(3(-4x) + 2)) - 2)/(1((-2(-4x) - 2)/(3(-4x) + 2)) + 1)) - 1| = 1(3|(-5x + 4) + 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": "Find the number of times the graphs of f(x) = 1(3cos(2π(1(5x^2 + 4x) + 5) - 3) - 2)^2 + 4(3cos(2π(1(5x^2 + 4x) + 5) - 3) - 2) - 2 and g(x) = -1(4((3x)/(2x - 1))^2 - 5((3x)/(2x - 1)) + 1)^2 - 4(4((3x)/(2x - 1))^2 - 5((3x)/(2x - 1)) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer... | 40 | MATH |
[
{
"content": "How many solutions does the equation 5(3|((-2(1x) + 3)/(-3(1x) - 1)) + 1| - 3) - 2 = -1|((-3(5x - 5) - 1)/(-1(5x - 5) - 1)) - 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(3(1(-4x^2 + 2x - 4)^2 + 5(-4x^2 + 2x - 4) + 2)^2 - 3) + 2)/(2(3(1(-4x^2 + 2x - 4)^2 + 5(-4x^2 + 2x - 4) + 2)^2 - 3) + 1) and g(x) = (3(1(-2|x| + 1) - 5) + 1)/(2(1(-2|x| + 1) - 5) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the a... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(2(-1|((-1x - 1)/(1x)) + 2|)^2 - 3(-1|((-1x - 1)/(1x)) + 2|) - 5) - 2)/(2(2(-1|((-1x - 1)/(1x)) + 2|)^2 - 3(-1|((-1x - 1)/(1x)) + 2|) - 5) + 2) and g(x) = 3(5(-3sin(1πx - 3) - 1) + 1)^2 + 2(5(-3sin(1πx - 3) - 1) + 1) - 5 wh... | 35 | MATH |
[
{
"content": "How many solutions does the equation 2|(3cos(2π(3(-2|x + 3| + 2) - 4)) + 3) - 1| - 2 = 5(3sin(3π(1|x - 3| - 2) - 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2((2(3sin(1π(3x + 5) + 3) - 2) + 3)/(1(3sin(1π(3x + 5) + 3) - 2) + 2)) - 4 and g(x) = 1(-4(1x + 3) + 2) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-1cos(2π(-4(3x - 3)^2 - 4(3x - 3) + 4) + 2) + 2)^2 + 1(-1cos(2π(-4(3x - 3)^2 - 4(3x - 3) + 4) + 2) + 2) - 1 and g(x) = 1|(5(2x - 2)) + 1| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 480 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-1|(3|(-3e^(1x) + 3) + 3| + 3) - 2| - 1) + 1| and g(x) = -3|(5((3x - 1)/(-2x + 3))^2 - 5((3x - 1)/(-2x + 3)) + 1) - 3| + 3 intersect 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(-1(-3(-3sin(2πx - 2) - 1) - 4)^2 + 5(-3(-3sin(2πx - 2) - 1) - 4) - 4) - 5 and g(x) = -1|(-2sin(3π(1|x| - 3) - 1)) - 2| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-2cos(2π(-3(3|x + 2| - 3)) - 2) - 2)^2 + 1(-2cos(2π(-3(3|x + 2| - 3)) - 2) - 2) + 2 and g(x) = 5((1(-3cos(2πx + 3) - 2) + 2)/(-2(-3cos(2πx + 3) - 2) + 3))^2 + 5((1(-3cos(2πx + 3) - 2) + 2)/(-2(-3cos(2πx + 3) - 2) + 3)) + 5 ... | 344 | MATH |
[
{
"content": "How many solutions does the equation (2(1sin(2π(1((3x)/(3x - 2)) - 3) - 3)) + 3)/(2(1sin(2π(1((3x)/(3x - 2)) - 3) - 3)) + 1) = -2e^(2((2(3|x| + 3) + 3)/(2(3|x| + 3) + 2))) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(2π((2(1(3x^2 + 4x - 2)^2 + 5(3x^2 + 4x - 2) - 2) + 2)/(-2(1(3x^2 + 4x - 2)^2 + 5(3x^2 + 4x - 2) - 2) + 3)) + 1) + 2 and g(x) = -1|((-1(2x - 5) - 2)/(2(2x - 5) - 1))| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \... | 26 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-3(-2(-1x^2 - 4x + 2) + 2)^2 - 3(-2(-1x^2 - 4x + 2) + 2) - 2) + 2| and g(x) = -3(-2(3|x + 3| + 2) + 5)^2 - 2(-2(3|x + 3| + 2) + 5) + 5 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) = 1|((-1(-4(-2sin(1πx - 1) - 1) + 4) - 1)/(-1(-4(-2sin(1πx - 1) - 1) + 4) + 2))| + 1 and g(x) = 3(-4(3e^(-1x) - 1) + 3) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 19 | MATH |
[
{
"content": "How many solutions does the equation (2((2(1(-1x + 5)^2 - 1(-1x + 5) - 5))/(-1(1(-1x + 5)^2 - 1(-1x + 5) - 5) + 2)) + 2)/(-2((2(1(-1x + 5)^2 - 1(-1x + 5) - 5))/(-1(1(-1x + 5)^2 - 1(-1x + 5) - 5) + 2)) + 3) = -5(2sin(3π(2x - 3) - 3) - 1) + 4 have for x in the interval [-10, 10]?\n\nPresent the answ... | 118 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(1sin(2π(1(-1x^2 + 1x + 5)^2 + 3(-1x^2 + 1x + 5) + 4) + 2) - 1) + 3)/(3(1sin(2π(1(-1x^2 + 1x + 5)^2 + 3(-1x^2 + 1x + 5) + 4) + 2) - 1) + 2) and g(x) = 4(-3sin(2π(-1x^2 - 2x + 3)) + 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in ... | 275 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(2(-2cos(2π(2x^2 - 5x - 5) - 2) + 3) - 1) - 3 and g(x) = -2((3(-1|x + 3| + 3) + 3)/(2(-1|x + 3| + 3) - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many solutions does the equation (3(2|(3sin(2π(2|x - 3|) + 2) - 2) - 3| - 1) - 2)/(1(2|(3sin(2π(2|x - 3|) + 2) - 2) - 3| - 1)) = 1|(-1cos(3π((-3x + 3)/(-2x + 3)) + 3) + 1) - 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 64 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-3sin(3π(3|((1x - 1)/(1x - 2)) + 1| + 1) - 2) + 2) + 2)/(-1(-3sin(3π(3|((1x - 1)/(1x - 2)) + 1| + 1) - 2) + 2) + 2) and g(x) = -2|(-1e^(-2((-1x - 3)/(3x))) + 3) - 1| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex f... | 37 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(2|(2cos(3π(-1|x + 2| - 1))) + 1| - 1) - 1)/(1(2|(2cos(3π(-1|x + 2| - 1))) + 1| - 1)) and g(x) = -3|(1|(-5x - 1) - 3| - 3)| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 31 | MATH |
[
{
"content": "How many solutions does the equation -2(2(1(-2cos(3πx + 2)) - 2)^2 + 2(1(-2cos(3πx + 2)) - 2) + 4) + 3 = -3|(4(5x^2 + 1x - 5)^2 - 4(5x^2 + 1x - 5) - 3) + 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1|(-1|((1x)/(2x - 1)) - 3| - 1) - 2| + 2)^2 + 2(1|(-1|((1x)/(2x - 1)) - 3| - 1) - 2| + 2) + 3 and g(x) = (-2(2sin(3π((2x - 3)/(1x + 2)) + 3) - 2) + 1)/(-2(2sin(3π((2x - 3)/(1x + 2)) + 3) - 2) - 2) intersect in the interval [-10, 10].\n\nPresent the a... | 11 | MATH |
[
{
"content": "How many solutions does the equation 4(1cos(1π(-3((3x + 2)/(-1x + 2)) + 2) + 1))^2 - 3(1cos(1π(-3((3x + 2)/(-1x + 2)) + 2) + 1)) - 4 = 2(1(-5x - 3)^2 - 5(-5x - 3) + 2)^2 - 3(1(-5x - 3)^2 - 5(-5x - 3) + 2) - 4 have for x 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) = -2(5(-3cos(2π(-1x + 2) - 3)) + 4) + 1 and g(x) = -1|(4(-1|x| + 3) - 3) - 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many solutions does the equation -3|(3|(-1|(-3cos(1πx - 1)) + 3| - 2) + 2| - 3) - 1| + 3 = -1((-1(3|x - 1| - 3) - 2)/(-2(3|x - 1| - 3) + 3)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(2|(2|(1e^(2x) + 3) - 3| - 3)| + 3) + 1 and g(x) = -3|(-2(-2sin(2πx + 3) - 2)^2 + 2(-2sin(2πx + 3) - 2) - 5) - 3| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 21 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-2(3(-1x^2 - 3x + 2))^2 + 5(3(-1x^2 - 3x + 2)) - 4) + 3| + 2 and g(x) = (-2(5(-2|x - 2| - 1) + 4) - 3)/(2(5(-2|x - 2| - 1) + 4) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3|(2(3e^(-2(-4x^2 + 4)) - 3)^2 - 4(3e^(-2(-4x^2 + 4)) - 3) - 2) + 2| + 2 = 3|((2(1cos(2πx + 3) - 1) + 3)/(-2(1cos(2πx + 3) - 1) - 1)) - 3| + 2 have for x 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) = (1(5((2(-2|x + 3| + 1) + 2)/(-1(-2|x + 3| + 1) + 3)) - 3))/(-1(5((2(-2|x + 3| + 1) + 2)/(-1(-2|x + 3| + 1) + 3)) - 3) - 1) and g(x) = (1(-5(2x)^2 - 1(2x) + 4) + 1)/(1(-5(2x)^2 - 1(2x) + 4)) intersect in the interval [-10, 10].\n\nPresent the answer in ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3(1(-3((-1x + 2)/(-1x - 2)) - 5)^2 - 4(-3((-1x + 2)/(-1x - 2)) - 5) + 5) = 3(3cos(3π((1x + 3)/(1x)) - 2) + 2)^2 + 1(3cos(3π((1x + 3)/(1x)) - 2) + 2) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 78 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(1|(1(-1|x - 1| + 1) + 3) + 1| - 2) - 1)/(-1(1|(1(-1|x - 1| + 1) + 3) + 1| - 2) - 2) and g(x) = (-1(4((-3x - 1)/(-2x))^2 + 2((-3x - 1)/(-2x)) - 1))/(2(4((-3x - 1)/(-2x))^2 + 2((-3x - 1)/(-2x)) - 1) + 1) intersect in the interval [-10, 10].\n\nPresent... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-2(-4(2|x - 1|) + 5)) + 1 and g(x) = 1|((-1(3x + 4) + 2)/(2(3x + 4) + 1)) + 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-4(-1(1x^2 + 3x - 4) + 2)^2 - 3(-1(1x^2 + 3x - 4) + 2) + 1)^2 + 1(-4(-1(1x^2 + 3x - 4) + 2)^2 - 3(-1(1x^2 + 3x - 4) + 2) + 1) - 4 and g(x) = (1((1((1x - 2)/(3x + 3)) + 2)/(3((1x - 2)/(3x + 3)) - 2)) - 1)/(-2((1((1x - 2)/(3x + 3))... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-3sin(2π((-2((-2x - 1)/(3x + 3)) + 2)/(-2((-2x - 1)/(3x + 3)) + 1)) - 2)) - 3| + 2 and g(x) = 1|((-2(-1x^2 + 2x + 2) - 2)/(2(-1x^2 + 2x + 2) + 3)) + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 21 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(2sin(2π(4(-3x^2 + 2x)^2 - 2(-3x^2 + 2x) + 4) - 1)) + 3| - 1 and g(x) = 5(3sin(3π(-1|x + 2| - 3)) + 3)^2 + 5(3sin(3π(-1|x + 2| - 3)) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 94 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2|(-1sin(1π((1x - 2)/(1x)) + 3) - 2) - 3| + 1) + 3)/(-2(2|(-1sin(1π((1x - 2)/(1x)) + 3) - 2) - 3| + 1) - 1) and g(x) = -1|(2sin(3π((2x - 3)/(-1x - 2)) + 1) - 2) + 2| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex form... | 223 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(1e^(2(-3(-2x + 2) - 3)) - 2) - 5 and g(x) = 5(1cos(1π(-1x^2 + 5x - 2) - 2) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 155 | MATH |
[
{
"content": "How many solutions does the equation -5(-2|(-1(2cos(2πx) + 1) + 3)| + 2)^2 + 5(-2|(-1(2cos(2πx) + 1) + 3)| + 2) + 4 = -5(2|(1|x + 2| + 2) + 1| + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many solutions does the equation (1(-2(2(-4x - 3)^2 - 5(-4x - 3) - 3) - 4) + 1)/(2(-2(2(-4x - 3)^2 - 5(-4x - 3) - 3) - 4) - 1) = 3sin(2π(-4((1x - 3)/(3x - 2)) + 5)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 95 | MATH |
[
{
"content": "How many solutions does the equation -2|((-3(2|(-4x^2 + 5x + 4) - 3| + 2))/(1(2|(-4x^2 + 5x + 4) - 3| + 2) + 3)) - 3| + 3 = -3|(3cos(3π(-1|x + 2| + 3) + 3)) + 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-4(-4((2x + 3)/(-3x - 2)) - 2) - 4) + 1)/(-1(-4(-4((2x + 3)/(-3x - 2)) - 2) - 4) + 2) and g(x) = -4(2(-3x + 1)^2 + 5(-3x + 1) - 5)^2 - 4(2(-3x + 1)^2 + 5(-3x + 1) - 5) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5((1(-1(-1x^2 + 5x - 5)^2 - 4(-1x^2 + 5x - 5) + 4) - 1)/(3(-1(-1x^2 + 5x - 5)^2 - 4(-1x^2 + 5x - 5) + 4) - 2))^2 + 5((1(-1(-1x^2 + 5x - 5)^2 - 4(-1x^2 + 5x - 5) + 4) - 1)/(3(-1(-1x^2 + 5x - 5)^2 - 4(-1x^2 + 5x - 5) + 4) - 2)... | 123 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(5(-1(-2|x - 3| + 3)^2 - 5(-2|x - 3| + 3) + 2)^2 + 4(-1(-2|x - 3| + 3)^2 - 5(-2|x - 3| + 3) + 2) - 5)^2 + 5(5(-1(-2|x - 3| + 3)^2 - 5(-2|x - 3| + 3) + 2)^2 + 4(-1(-2|x - 3| + 3)^2 - 5(-2|x - 3| + 3) + 2) - 5) - 1 and g(x) =... | 20 | MATH |
[
{
"content": "How many solutions does the equation -5(1((1((-2x)/(-2x + 1)) - 2)/(-2((-2x)/(-2x + 1)) + 3))^2 + 5((1((-2x)/(-2x + 1)) - 2)/(-2((-2x)/(-2x + 1)) + 3)) + 5) - 3 = (2(2|(1|x - 1|)| - 2) + 1)/(3(2|(1|x - 1|)| - 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-3(-3cos(2π(-2x - 1) + 2) - 2) + 3)/(-3(-3cos(2π(-2x - 1) + 2) - 2))) - 2 and g(x) = 2cos(2π((-3(-1x - 3) - 1)/(-1(-1x - 3) - 1)) - 2) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 81 | MATH |
[
{
"content": "How many solutions does the equation (-2(3(-3(2x^2 + 5x - 1)^2 - 4(2x^2 + 5x - 1) + 5)^2 + 3(-3(2x^2 + 5x - 1)^2 - 4(2x^2 + 5x - 1) + 5) - 1) + 3)/(2(3(-3(2x^2 + 5x - 1)^2 - 4(2x^2 + 5x - 1) + 5)^2 + 3(-3(2x^2 + 5x - 1)^2 - 4(2x^2 + 5x - 1) + 5) - 1) + 3) = -4((-3(3x^2 - 4x) - 2)/(1(3x^2 - 4x) - 3... | 5 | MATH |
[
{
"content": "How many solutions does the equation (-2(-1((-1(5x - 4) - 3)/(-1(5x - 4) + 3))^2 - 3((-1(5x - 4) - 3)/(-1(5x - 4) + 3)) - 1) + 2)/(-1(-1((-1(5x - 4) - 3)/(-1(5x - 4) + 3))^2 - 3((-1(5x - 4) - 3)/(-1(5x - 4) + 3)) - 1) + 3) = -2|((2(3x^2 + 1x - 4) - 1)/(1(3x^2 + 1x - 4))) - 1| + 2 have for x in the... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(3(1cos(2π((-3x - 1)/(1x - 2)) + 2) + 2)^2 - 4(1cos(2π((-3x - 1)/(1x - 2)) + 2) + 2)) - 1 and g(x) = 1e^(1(-5(-1|x - 3|) - 3)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 35 | MATH |
[
{
"content": "How many solutions does the equation -2(1(4(-3sin(3πx - 1) + 3)^2 + 5(-3sin(3πx - 1) + 3) - 2)) = 1(-1cos(1π((2x + 3)/(2x + 1)) + 3))^2 + 5(-1cos(1π((2x + 3)/(2x + 1)) + 3)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 66 | MATH |
[
{
"content": "How many solutions does the equation (-3(3(-2((2x)/(-1x + 2))^2 + 5((2x)/(-1x + 2)) - 2)^2 - 5(-2((2x)/(-1x + 2))^2 + 5((2x)/(-1x + 2)) - 2) - 1) + 2)/(1(3(-2((2x)/(-1x + 2))^2 + 5((2x)/(-1x + 2)) - 2)^2 - 5(-2((2x)/(-1x + 2))^2 + 5((2x)/(-1x + 2)) - 2) - 1) - 3) = -2|(-2(-5x - 2) + 5) + 3| + 1 ha... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(-3((3((-2x)/(2x + 3)))/(-3((-2x)/(2x + 3)) + 2)) + 5) - 2| + 1 and g(x) = -3sin(1π((-3(3x + 3) + 2)/(-2(3x + 3) + 3)) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4((2(-3sin(2π(1x^2 - 3x + 1)) - 1) - 3)/(-1(-3sin(2π(1x^2 - 3x + 1)) - 1) + 2)) - 1 and g(x) = (-2(1(1x^2 - 3x - 4)^2 - 1(1x^2 - 3x - 4) + 3) + 2)/(-1(1(1x^2 - 3x - 4)^2 - 1(1x^2 - 3x - 4) + 3) + 2) intersect in the interval [-10, 10].\n\nPresent the ... | 410 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-3(-2(-3sin(3πx + 1) + 2)^2 + 1(-3sin(3πx + 1) + 2) + 2)^2 + 5(-2(-3sin(3πx + 1) + 2)^2 + 1(-3sin(3πx + 1) + 2) + 2))^2 - 3(-3(-2(-3sin(3πx + 1) + 2)^2 + 1(-3sin(3πx + 1) + 2) + 2)^2 + 5(-2(-3sin(3πx + 1) + 2)^2 + 1(-3sin(... | 358 | MATH |
[
{
"content": "How many solutions does the equation (3((3(-1((3x - 3)/(-3x + 4)) - 5) + 2)/(-3(-1((3x - 3)/(-3x + 4)) - 5) - 2)) + 2)/(-2((3(-1((3x - 3)/(-3x + 4)) - 5) + 2)/(-3(-1((3x - 3)/(-3x + 4)) - 5) - 2))) = 3sin(3π(-3(-1|x - 3| + 1)^2 + 2(-1|x - 3| + 1) - 1)) - 2 have for x in the interval [-10, 10]?\n\n... | 1226 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4((2((2(-1cos(1πx - 2) - 2))/(1(-1cos(1πx - 2) - 2) - 3)) - 2)/(2((2(-1cos(1πx - 2) - 2))/(1(-1cos(1πx - 2) - 2) - 3)) + 2)) - 1 and g(x) = (2(-2(-2sin(2πx + 2) + 2) + 5) - 3)/(2(-2(-2sin(2πx + 2) + 2) + 5) - 1) intersect in the interval [-10, 10].\n\n... | 40 | MATH |
[
{
"content": "How many solutions does the equation (2(-3e^(1((-3(1|x - 1| - 2) + 2)/(-3(1|x - 1| - 2) - 1)))) - 1)/(1(-3e^(1((-3(1|x - 1| - 2) + 2)/(-3(1|x - 1| - 2) - 1)))) - 3) = 1(1cos(1π((1x)/(1x - 2)) - 2) + 2)^2 - 4(1cos(1π((1x)/(1x - 2)) - 2) + 2) + 4 have for x in the interval [-10, 10]?\n\nPresent the ... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2(1(1|(-2sin(2πx + 3) - 1) + 2|)^2 + 2(1|(-2sin(2πx + 3) - 1) + 2|) - 5) + 1)/(2(1(1|(-2sin(2πx + 3) - 1) + 2|)^2 + 2(1|(-2sin(2πx + 3) - 1) + 2|) - 5) + 2) = -5(2(1cos(1πx + 1)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format... | 29 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1cos(2π(-5((-1x)/(3x + 1))^2 + 5((-1x)/(3x + 1)) - 5) + 1) - 3) - 5 and g(x) = 4(-2cos(1π((3x - 1)/(1x + 3)) - 3) - 3)^2 + 5(-2cos(1π((3x - 1)/(1x + 3)) - 3) - 3) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex for... | 102 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(2e^(2((1(-4x^2 - 3x))/(1(-4x^2 - 3x) + 1))) - 3) + 2)/(-2(2e^(2((1(-4x^2 - 3x))/(1(-4x^2 - 3x) + 1))) - 3) + 3) and g(x) = 5(3|(2sin(2πx))| - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(2sin(2π(1(2x^2 + 1x - 1)^2 + 2(2x^2 + 1x - 1) - 5) - 3) + 2) + 1)/(1(2sin(2π(1(2x^2 + 1x - 1)^2 + 2(2x^2 + 1x - 1) - 5) - 3) + 2) - 2) and g(x) = -1|(-3sin(1π((-1x)/(-1x + 2)) + 3)) - 1| in the range -10 ≤ x ≤ 10?\n\nPresent th... | 187 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|((-3(-3(3|x - 3| + 2)^2 + 1(3|x - 3| + 2) + 3) + 1)/(3(-3(3|x - 3| + 2)^2 + 1(3|x - 3| + 2) + 3) - 2)) - 3| + 1 and g(x) = (2((-2(-3sin(3πx - 2) + 2) - 2)/(1(-3sin(3πx - 2) + 2) + 3)) - 3)/(3((-2(-3sin(3πx - 2) + 2) - 2)/(... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((-1(1(3cos(3πx - 3) + 1)) - 1)/(-3(1(3cos(3πx - 3) + 1)) + 4)) + 4 and g(x) = -1|(3cos(1π(-4x - 1) - 3) - 2)| - 2 in the range -10 ≤ x ≤ 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) = 3((-2(-4(1x - 2) + 1) - 2)/(-3(-4(1x - 2) + 1) + 4))^2 - 1((-2(-4(1x - 2) + 1) - 2)/(-3(-4(1x - 2) + 1) + 4)) + 1 and g(x) = 2|(3sin(1π((2x)/(-2x + 1)) - 2) - 2)| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format... | 24 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((-2(-5(-2|x - 3|) - 1) + 2)/(-3(-5(-2|x - 3|) - 1)))^2 + 4((-2(-5(-2|x - 3|) - 1) + 2)/(-3(-5(-2|x - 3|) - 1))) + 5 and g(x) = -1|(-3(3|x - 2|) - 1) + 3| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\box... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-4(3|((-1x)/(3x - 2)) - 3| - 2) + 1) - 3 and g(x) = -1(-1|(3x^2 + 4x - 2) + 1| + 2) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1(-5(1x + 3)^2 - 4(1x + 3) - 3) + 4)^2 - 3(1(-5(1x + 3)^2 - 4(1x + 3) - 3) + 4) and g(x) = -3cos(1π(-1(-2x - 3) + 4) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation -4(1(-2|((2x + 1)/(-3x - 2)) + 3| + 1)^2 + 5(-2|((2x + 1)/(-3x - 2)) + 3| + 1) - 1) - 1 = -1|(2|(3|x + 2| + 1) + 3| + 1) - 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(1|(2cos(2π(-5x - 2) - 1)) - 2| + 1) - 1)/(1(1|(2cos(2π(-5x - 2) - 1)) - 2| + 1)) and g(x) = 2|(-1(3|x + 2|)^2 - 1(3|x + 2|) + 4)| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2|(-3|(-3sin(1π(-1x^2 + 2x - 1) - 3) + 1) - 1| - 3)| + 1 = -2(1e^(-1(-1x^2 + 3x + 5)) - 1)^2 + 4(1e^(-1(-1x^2 + 3x + 5)) - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(3π(1(2|(4x) + 1| - 2)^2 + 1(2|(4x) + 1| - 2) + 3) - 3) - 2 and g(x) = 3cos(3π(1|(2x^2 + 3x - 4) + 2| - 1) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 810 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(2π(-2|(2(-1x^2 + 3x)^2 + 3(-1x^2 + 3x) + 1) + 2|) + 2) - 2 and g(x) = 1e^(-1(4(-3x^2 + 2x + 3)^2 + 5(-3x^2 + 2x + 3) + 3)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 1106 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(2|((2(1|x + 1| - 2))/(-3(1|x + 1| - 2) + 4)) + 3|) + 2| + 2 and g(x) = -3cos(2π(-1|(1x^2 - 5x - 5) + 3| - 2) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(2(-4(-5x + 2) - 3) + 4) + 2)/(2(2(-4(-5x + 2) - 3) + 4)) and g(x) = -1|(3cos(2π(-1x + 1) - 2) + 2) - 1| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 79 | MATH |
[
{
"content": "How many solutions does the equation (-2(1((-2(4x^2 - 3x + 1) - 1)/(-2(4x^2 - 3x + 1) - 3)) + 5) - 1)/(2(1((-2(4x^2 - 3x + 1) - 1)/(-2(4x^2 - 3x + 1) - 3)) + 5) - 1) = -3cos(3π(2(3|x - 3| + 1)) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 360 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(3(-1(-1|x + 1| + 3))^2 + 5(-1(-1|x + 1| + 3)) - 5) - 4 and g(x) = 1|((-1(-4x - 2))/(3(-4x - 2) - 2)) - 2| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation -3|((3(-3(-1x - 3)^2 - 3(-1x - 3) + 5) - 1)/(3(-3(-1x - 3)^2 - 3(-1x - 3) + 5))) - 2| + 2 = 1sin(3π((1((1x + 2)/(-3x - 2)) + 2)/(-2((1x + 2)/(-3x - 2)) + 1)) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(2π((2(1(-5x)^2 - 2(-5x)) + 1)/(2(1(-5x)^2 - 2(-5x)) + 3)) + 3) + 3 and g(x) = -2e^(-1(2|(-2x^2 - 4x - 4) - 1| - 1)) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-1((-3(3cos(2πx - 2) + 1) - 2)/(2(3cos(2πx - 2) + 1) + 2)) + 1) - 5 and g(x) = -2|((1((1x + 2)/(2x - 1)) - 3)/(3((1x + 2)/(2x - 1)) - 2)) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 35 | MATH |
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