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) = 4(2|(-3sin(1πx - 2) - 2) + 2| - 2) + 1 and g(x) = -1(2e^(-2x) - 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1sin(2π(-3(1(3|x - 2| + 3) + 1)^2 + 2(1(3|x - 2| + 3) + 1) - 1) - 3) + 1 and g(x) = 1|(-2(-5x^2 + 4x + 2) + 3) + 3| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(3π((2(-1|(2x^2 + 2x - 4)| + 1) - 2)/(3(-1|(2x^2 + 2x - 4)| + 1) - 2)) - 1) + 3 and g(x) = -1(-1|(-1x^2 + 4x - 4) + 3| - 2) - 1 intersect 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|(4((-2(3cos(2πx - 2) - 3) + 1)/(-1(3cos(2πx - 2) - 3) - 3))) + 3| - 1 and g(x) = (-2(2(-3e^(-1x) + 2) - 4))/(-1(2(-3e^(-1x) + 2) - 4)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 40 | MATH |
[
{
"content": "How many solutions does the equation 3sin(3π((2(5(-1x - 3)^2 - 3(-1x - 3) - 3) - 2)/(2(5(-1x - 3)^2 - 3(-1x - 3) - 3) - 1))) - 3 = (-2(2|(5x^2 + 3x + 1)| + 3) - 2)/(2(2|(5x^2 + 3x + 1)| + 3) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 28 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-1|(4(-3sin(2πx - 3) + 2)^2 - 2(-3sin(2πx - 3) + 2) - 3) + 3| - 2) - 5 and g(x) = 3(-3|(3x + 5) - 2| + 2)^2 - 1(-3|(3x + 5) - 2| + 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 8 | MATH |
[
{
"content": "How many solutions does the equation 2sin(1π(-1|(-3x + 3) + 2| + 2) + 3) + 3 = (2(3cos(2πx + 3) - 1) + 3)/(2(3cos(2πx + 3) - 1) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(3|(2((-2x + 3)/(1x + 2)) + 2)|)^2 + 4(3|(2((-2x + 3)/(1x + 2)) + 2)|) - 3 and g(x) = -3sin(3π(2(-1x^2 - 2x + 4) - 1) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 93 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-5(-2|x| + 2)^2 + 1(-2|x| + 2) + 3) + 1)/(-1(-5(-2|x| + 2)^2 + 1(-2|x| + 2) + 3) + 2) and g(x) = 2cos(2π((-3x + 2)/(3x + 3)) - 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(3((-1(2|x + 3| + 1) - 2)/(-1(2|x + 3| + 1) + 2))^2 - 1((-1(2|x + 3| + 1) - 2)/(-1(2|x + 3| + 1) + 2)) + 2) + 2 and g(x) = 3|(3cos(2π(-3x^2 - 3x - 4) + 2)) - 3| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: ... | 670 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((3(-3|(-4x^2 + 4x - 2) + 2| + 3))/(-1(-3|(-4x^2 + 4x - 2) + 2| + 3))) + 2 and g(x) = (-1(-1|(1sin(2πx - 3) + 3) - 2| + 2) - 3)/(-2(-1|(1sin(2πx - 3) + 3) - 2| + 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\bo... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(2sin(2π(3x - 3) - 2) - 1) - 2)/(3(2sin(2π(3x - 3) - 2) - 1) + 3) and g(x) = (-2(1sin(3πx + 1) + 1) + 2)/(-1(1sin(3πx + 1) + 1) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 101 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(1|(1(-3x^2 + 3x - 4) + 5) - 2| - 2) - 1)/(3(1|(1(-3x^2 + 3x - 4) + 5) - 2| - 2) + 3) and g(x) = 2(-3|(1x^2 + 4x - 4) - 3|) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(1(3sin(2πx - 1) - 1) + 3) + 1)/(-2(1(3sin(2πx - 1) - 1) + 3) - 2) and g(x) = 2cos(1π(5x + 2) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(3π((-3(-3|(-4x^2 + 2x - 2) + 3| - 3) + 2)/(-1(-3|(-4x^2 + 2x - 2) + 3| - 3) + 3)) - 2) + 1 and g(x) = (-2(-5(-1x^2 - 5x)))/(-1(-5(-1x^2 - 5x)) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{You... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1(3e^(-1(-3|x - 2| - 2)) + 2)^2 + 2) - 1)/(-1(-1(3e^(-1(-3|x - 2| - 2)) + 2)^2 + 2) - 2) and g(x) = 1cos(1π(3|(4x^2 + 1x + 2) - 2| - 2)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 849 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-1cos(3π((1(-2x^2 - 4x + 5) + 3)/(-3(-2x^2 - 4x + 5) + 1))) + 2) + 1 and g(x) = -1(2|(1sin(1πx - 3) + 3) - 3|) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(2π(4(-2x^2 - 5x - 3) - 3) + 2) - 1 and g(x) = 3e^(2(-1x - 2)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 858 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(2|((-2(2cos(1πx - 3) + 3) + 2)/(-1(2cos(1πx - 3) + 3) + 2)) + 1| - 1) + 4 and g(x) = -4(-2e^(-1((2x + 1)/(2x - 3))) - 1)^2 - 4(-2e^(-1((2x + 1)/(2x - 3))) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\b... | 37 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-3|(2sin(3πx) + 3) - 1| + 2)^2 - 1(-3|(2sin(3πx) + 3) - 1| + 2) + 4 and g(x) = -4(1e^(2x) - 3) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-2|(3sin(1πx + 3) + 1) - 2| - 2) - 1 and g(x) = -1(1|x - 3| - 1)^2 - 4(1|x - 3| - 1) + 4 intersect 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(1π(3|(2x - 5) - 2| + 1) - 1) - 2 and g(x) = -1sin(3π(2x + 1) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 82 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(2(1|(-4x + 5) + 3|)^2 + 1(1|(-4x + 5) + 3|) + 2) - 1)/(-1(2(1|(-4x + 5) + 3|)^2 + 1(1|(-4x + 5) + 3|) + 2)) and g(x) = 2|(4(3sin(1πx) - 1)^2 - 4) + 2| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxe... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-2|(3x + 4) - 3| + 3) - 2)/(-2(-2|(3x + 4) - 3| + 3) - 3) and g(x) = (1(-2sin(3πx - 3) - 2))/(-3(-2sin(3πx - 3) - 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(1(-4((-1x + 3)/(-3x - 1))^2 - 4((-1x + 3)/(-3x - 1)) + 2) + 3) - 1| - 2 and g(x) = -3(2(2sin(2πx - 3) - 1) - 3)^2 + 5(2(2sin(2πx - 3) - 1) - 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 38 | MATH |
[
{
"content": "How many solutions does the equation -2(-3|((3(2sin(2πx - 2) + 2) - 1)/(-2(2sin(2πx - 2) + 2) + 3)) - 1|) - 2 = -1e^(2(-3((2x)/(3x - 1))^2 + 2)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 50 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1e^(-2(-2((2x - 1)/(-2x - 2))^2 - 4((2x - 1)/(-2x - 2)) + 2)) - 3 and g(x) = 3cos(3π(-5x^2 + 5x - 1) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1497 | MATH |
[
{
"content": "How many solutions does the equation 2|((2(2cos(2πx + 1) - 1))/(-3(2cos(2πx + 1) - 1) + 2)) + 2| = -3((2x + 2)/(-1x - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(1|(-1cos(2π(1x - 3) + 1)) - 2| - 2) - 1)/(-2(1|(-1cos(2π(1x - 3) + 1)) - 2| - 2) + 1) and g(x) = 1(-4(-1sin(2πx - 2))^2 - 5(-1sin(2πx - 2)) - 5) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your a... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((-2(3cos(3π(3x^2 + 1x + 5) - 3) - 2) - 3)/(-1(3cos(3π(3x^2 + 1x + 5) - 3) - 2) - 2)) - 1 and g(x) = 1|(-4((1x - 1)/(3x + 3))^2 - 2((1x - 1)/(3x + 3)) - 1)| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\b... | 350 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-2cos(1π((3(-4x^2 - 2x + 1) + 2)/(-2(-4x^2 - 2x + 1))) - 3)) + 2| and g(x) = (-1(3|(2x + 4) - 1| + 2) - 2)/(2(3|(2x + 4) - 1| + 2) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2cos(3π(-3x) + 2) + 2) - 1)/(3(2cos(3π(-3x) + 2) + 2) - 2) and g(x) = -1(-2|x + 1|)^2 - 5(-2|x + 1|) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 45 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(2(1e^(1(-5x + 2)) + 3)^2 + 1(1e^(1(-5x + 2)) + 3)))/(-3(2(1e^(1(-5x + 2)) + 3)^2 + 1(1e^(1(-5x + 2)) + 3)) - 1) and g(x) = 1|(2cos(1π(-3x^2 + 4x - 1) - 2)) + 1| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: ... | 966 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-2((-3x + 1)/(-1x - 3)) + 3) - 3| + 3 and g(x) = 3cos(2π(-2x^2 - 3x - 5) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 200 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(3sin(3π(4(3x + 4)^2 + 5(3x + 4) + 5) - 1)) + 3)/(-2(3sin(3π(4(3x + 4)^2 + 5(3x + 4) + 5) - 1)) + 3) and g(x) = 3cos(2π((1(3|x + 1| + 3) - 3)/(2(3|x + 1| + 3) - 1)) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: ... | 318 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-2sin(1π(-3|x - 3| + 3) - 1) + 2)^2 - 1(-2sin(1π(-3|x - 3| + 3) - 1) + 2) + 3 and g(x) = 1|(-3x + 4) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(2(-2e^(-2(-3|x - 3| - 2)) - 3) - 4) - 2)/(-3(2(-2e^(-2(-3|x - 3| - 2)) - 3) - 4) + 1) and g(x) = 2|(-1(2sin(1πx + 2) + 1) + 3) - 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(1(-2sin(3πx + 1) - 2)) + 2)/(-3(1(-2sin(3πx + 1) - 2)) - 2) and g(x) = -5(-3|x + 2|) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(1π(2|((-2x - 1)/(3x + 3)) + 1| + 3) - 1) + 1 and g(x) = 4(-5x - 5)^2 - 1(-5x - 5) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4((1(1(-1|x + 1| + 3)^2 - 1(-1|x + 1| + 3)) - 2)/(-2(1(-1|x + 1| + 3)^2 - 1(-1|x + 1| + 3)) + 3)) and g(x) = -4(-3|(-2sin(3πx)) - 2| + 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 60 | MATH |
[
{
"content": "How many solutions does the equation 4(-1sin(1π((3(1x^2 - 2x - 3) + 3)/(2(1x^2 - 2x - 3) + 3)) + 3) + 2) = 2(3|((-1x + 1)/(1x + 3)) - 2| - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "How many solutions does the equation 1(-2|((-1(-3sin(3πx - 3) + 1) + 3)/(-2(-3sin(3πx - 3) + 1) + 1)) - 1| + 3) - 2 = (-1(1(-1|x - 1|)^2 + 2(-1|x - 1|) - 4) + 2)/(-2(1(-1|x - 1|)^2 + 2(-1|x - 1|) - 4) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 60 | MATH |
[
{
"content": "How many solutions does the equation 3|(2(3((-1x - 3)/(-1x + 2)) - 2)^2 - 1(3((-1x - 3)/(-1x + 2)) - 2) + 2) + 2| - 3 = 3|((1(-2cos(2πx) + 3) - 1)/(-3(-2cos(2πx) + 3) + 4)) - 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 30 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(1|(-3x^2 + 4x - 2) - 1| - 3) + 2)/(-2(1|(-3x^2 + 4x - 2) - 1| - 3) - 2) and g(x) = 3(1cos(3πx - 2))^2 - 2(1cos(3πx - 2)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(1π(-1|(1x^2 + 3x + 4) - 3| - 1) + 2) - 3 and g(x) = 3e^(-1(4x + 3)) - 2 intersect 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) = (-1(2|(-5x^2 + 4x - 5)|) + 1)/(-3(2|(-5x^2 + 4x - 5)|) + 2) and g(x) = 1(3cos(1πx + 2))^2 - 5(3cos(1πx + 2)) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(1π(1(2|x - 3| + 3)^2 - 5(2|x - 3| + 3) - 4) - 3) + 2 and g(x) = 2|((-3x + 1)/(-2x - 3)) + 3| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many solutions does the equation -1(-2e^(1(-1|x - 3| - 1)) - 3) + 3 = 1(3sin(1πx - 3) + 1)^2 + 4(3sin(1πx - 3) + 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1(3sin(1πx + 2) + 1) - 5) - 3)/(2(-1(3sin(1πx + 2) + 1) - 5) + 2) and g(x) = 2sin(2π(3x - 4) - 3) + 3 intersect 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) = -1sin(1π(-5(1x - 3)^2 - 2(1x - 3) - 1) + 3) - 2 and g(x) = 2((-1x - 2)/(-3x + 2)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 68 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((-2(-1(-2x - 5)^2 + 4(-2x - 5) - 2) - 3)/(2(-1(-2x - 5)^2 + 4(-2x - 5) - 2))) - 1| + 3 and g(x) = -5(-2|(-1cos(3πx - 1) - 2) + 1| - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|((-3(3sin(2πx - 3) - 3))/(1(3sin(2πx - 3) - 3) + 2)) - 1| and g(x) = 3cos(1π(-1x^2 - 5x - 4) - 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 144 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(3cos(3π(-4(-1x - 1)^2 + 4(-1x - 1) + 2) + 2) + 1) - 2)/(3(3cos(3π(-4(-1x - 1)^2 + 4(-1x - 1) + 2) + 2) + 1) - 1) and g(x) = -2|(-4(1cos(1πx - 2) + 3)^2 + 5(1cos(1πx - 2) + 3) + 1) + 1| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent t... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(3|(2cos(3π(-5x^2 - 3x + 2) + 3) - 3) + 1| - 1) + 3 and g(x) = 2(1|(-1sin(2πx - 3) + 1) - 1| - 1)^2 - 4(1|(-1sin(2πx - 3) + 1) - 1| - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 778 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2e^(-1((-3(3(-1|x - 1| - 3)^2 + 2(-1|x - 1| - 3) + 1) - 1)/(3(3(-1|x - 1| - 3)^2 + 2(-1|x - 1| - 3) + 1) - 1))) + 3 and g(x) = -2(-2sin(3π(-3x - 1) - 2))^2 - 5(-2sin(3π(-3x - 1) - 2)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the ... | 180 | MATH |
[
{
"content": "How many solutions does the equation 3sin(3π(-2((2(-3|x - 1| - 2) - 2)/(3(-3|x - 1| - 2) + 2)) - 2) - 3) - 2 = -3|(2sin(1π(-5x^2 + 2x) + 2) - 1) - 2| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation -2((-3(-2cos(3πx - 1) + 3) - 3)/(2(-2cos(3πx - 1) + 3) + 3))^2 - 1((-3(-2cos(3πx - 1) + 3) - 3)/(2(-2cos(3πx - 1) + 3) + 3)) + 2 = -3cos(3π(-2|x + 2|) - 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 84 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(2π((2(2|x| - 1) + 1)/(-1(2|x| - 1) + 2)) + 2) + 1 and g(x) = (-3(-1x + 2))/(3(-1x + 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-1sin(3π(-5(-3|x - 2|)^2 + 2(-3|x - 2|) - 1) - 1) - 1) - 3 and g(x) = -3((-2(-1|x + 3| + 1) + 1)/(2(-1|x + 3| + 1) - 3))^2 + 1((-2(-1|x + 3| + 1) + 1)/(2(-1|x + 3| + 1) - 3)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex fo... | 249 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-5(2cos(3π(-5x^2 - 1x + 3) + 1) + 2) + 2) - 3)/(-3(-5(2cos(3π(-5x^2 - 1x + 3) + 1) + 2) + 2) - 3) and g(x) = -2|(4(-2sin(2πx + 2) - 2) + 4) - 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your... | 67 | MATH |
[
{
"content": "How many solutions does the equation 3cos(3π(-3(3|x - 2| - 1)^2 - 2)) + 1 = (2(1x) + 3)/(1(1x) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1410 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(3sin(3π(-2|x - 2| - 2) + 1) - 2) - 3)/(1(3sin(3π(-2|x - 2| - 2) + 1) - 2) + 3) and g(x) = (-2(-2e^(-2x) + 3) + 3)/(1(-2e^(-2x) + 3)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 118 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(-1(-2(-2|x| - 3)^2 - 2(-2|x| - 3) + 2) + 1) + 1)/(1(-1(-2(-2|x| - 3)^2 - 2(-2|x| - 3) + 2) + 1) - 2) and g(x) = 2|((1(3cos(1πx + 3) - 1))/(1(3cos(1πx + 3) - 1) + 1)) - 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in L... | 8 | MATH |
[
{
"content": "How many solutions does the equation -2(-3|(-2cos(1π((1x + 3)/(1x - 3)) - 3) + 1) - 1| - 1) - 3 = 2(-1|((-1x + 3)/(1x + 3)) - 2| + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 122 | MATH |
[
{
"content": "How many solutions does the equation -2(3cos(2π(-1|(-3x^2 + 3) + 3| + 2) - 2) + 3) + 1 = 1(-1((2x + 3)/(-2x - 3)) + 2)^2 - 2(-1((2x + 3)/(-2x - 3)) + 2) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1008 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-1(3|x| - 2)^2 - 5(3|x| - 2) + 5) + 3)/(2(-1(3|x| - 2)^2 - 5(3|x| - 2) + 5) - 3) and g(x) = 2|(-1x^2 + 4x + 3) - 3| - 3 in the range -10 ≤ x ≤ 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) = 2sin(3π((-1(-2|(4x + 5) - 3| - 1) + 1)/(-1(-2|(4x + 5) - 3| - 1) - 3)) - 3) - 3 and g(x) = -1e^(2((2(5x^2 - 5x - 2) + 2)/(-1(5x^2 - 5x - 2) + 2))) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your a... | 61 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1sin(3π(1|(1((1x + 3)/(-3x + 4))^2 + 2((1x + 3)/(-3x + 4)) + 1) + 3| + 1) + 1) - 3 and g(x) = 2|(1((2x + 2)/(1x - 3)) + 5)| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-1cos(3π(1|x + 2| + 2) + 2) + 2) + 3)/(-3(-1cos(3π(1|x + 2| + 2) + 2) + 2) - 2) and g(x) = -2sin(2π(2|x + 2| + 2) - 3) + 1 where -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) = -4(-2cos(1π(-1((-2x + 1)/(-3x + 4)) + 1) - 3) - 2)^2 - 4(-2cos(1π(-1((-2x + 1)/(-3x + 4)) + 1) - 3) - 2) - 1 and g(x) = (2(-3e^(-1(3|x|)) + 2) - 3)/(-2(-3e^(-1(3|x|)) + 2) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaT... | 32 | MATH |
[
{
"content": "How many solutions does the equation (-2(-4(1x + 1)^2 - 1(1x + 1) + 3) - 1)/(-1(-4(1x + 1)^2 - 1(1x + 1) + 3) + 2) = (-2(-3sin(3πx - 1) - 1) + 3)/(1(-3sin(3πx - 1) - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 61 | MATH |
[
{
"content": "How many solutions does the equation 1|(1((1x - 2)/(3x - 2))^2 - 3((1x - 2)/(3x - 2)) - 4) - 2| - 2 = (-1(-3cos(1πx - 2)) + 1)/(-3(-3cos(1πx - 2)) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many solutions does the equation (-2(-4(-2(-2sin(3πx - 2) + 1) - 1)^2 - 4(-2(-2sin(3πx - 2) + 1) - 1) - 4) - 1)/(2(-4(-2(-2sin(3πx - 2) + 1) - 1)^2 - 4(-2(-2sin(3πx - 2) + 1) - 1) - 4) - 1) = -4(-3sin(2π((-1x + 3)/(3x - 2)) + 3) - 3)^2 + 3(-3sin(2π((-1x + 3)/(3x - 2)) + 3) - 3) + 2 have for x ... | 36 | MATH |
[
{
"content": "How many solutions does the equation -2|((3(-2cos(1π(5x)) - 2) + 3)/(-1(-2cos(1π(5x)) - 2))) - 1| + 2 = -2(3cos(1π(1x^2 + 1x - 5) + 1) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 204 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3((2(-3|(-1cos(3πx - 1) - 1)| + 2) + 1)/(3(-3|(-1cos(3πx - 1) - 1)| + 2)))^2 - 1((2(-3|(-1cos(3πx - 1) - 1)| + 2) + 1)/(3(-3|(-1cos(3πx - 1) - 1)| + 2))) - 4 and g(x) = -5(-1e^(-2((2x - 3)/(-1x + 2))) - 1)^2 - 1(-1e^(-2((2x ... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(3(3cos(2πx + 3) - 1) + 3) - 2| - 3 and g(x) = -2cos(2π((3x - 2)/(-1x - 1)) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 86 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(1π((3(3|(1x^2 + 1x - 3) + 2|) - 1)/(-1(3|(1x^2 + 1x - 3) + 2|))) - 2) - 3 and g(x) = -1(1sin(3π((1x - 2)/(-1x + 2)) - 3) + 2)^2 - 1(1sin(3π((1x - 2)/(-1x + 2)) - 3) + 2) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in... | 29 | MATH |
[
{
"content": "How many solutions does the equation 4((2(-1(-1|x - 2| + 1)^2 + 2(-1|x - 2| + 1) + 2) - 2)/(3(-1(-1|x - 2| + 1)^2 + 2(-1|x - 2| + 1) + 2))) - 5 = 1sin(2π(-3((-2x - 2)/(2x - 3))^2 + 4((-2x - 2)/(2x - 3)) - 4) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3e^(-1(5(3|x| - 1)^2 + 3(3|x| - 1))) + 2 and g(x) = -3cos(2π((3x + 2)/(2x))) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(2π(5(1|x - 1| + 2) + 1) + 2) and g(x) = 2(-1|x - 3| + 2) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(2π(-3(-3|(-1x + 5) + 3| + 3)^2 + 5(-3|(-1x + 5) + 3| + 3) + 2) + 1) - 1 and g(x) = 1cos(2π(-4((-3x - 3)/(1x)) - 3) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 1454 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-1((-1x - 3)/(-2x + 3))^2 - 2((-1x - 3)/(-2x + 3)) - 4) + 3| and g(x) = -2sin(1π(-4x - 1) - 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 43 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5((-1(3|(-4x + 2) - 3| - 1) + 1)/(2(3|(-4x + 2) - 3| - 1) - 1))^2 - 1((-1(3|(-4x + 2) - 3| - 1) + 1)/(2(3|(-4x + 2) - 3| - 1) - 1)) + 4 and g(x) = 5(-2|(2x - 5) - 1|)^2 + 1(-2|(2x - 5) - 1|) + 5 in the range -10 ≤ x ≤ 10?\n\nPresen... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(1π(2(-2x - 2)^2 + 3(-2x - 2) - 2) + 1) - 3 and g(x) = 3|(-2sin(3πx - 1) + 3) - 3| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 129 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(2π((3(-2|(-1x - 5) - 3| + 3) - 3)/(1(-2|(-1x - 5) - 3| + 3) + 1)) + 2) and g(x) = -4(3cos(3π(2|x + 2| + 1) - 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 128 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(3π(-2|(5((-1x - 2)/(1x + 3)) - 4) + 3| - 2) - 2) + 2 and g(x) = -1cos(1π(-1(-2|x + 1| + 1)^2 + 4(-2|x + 1| + 1) + 4) + 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 563 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(3π(-3|((-3x)/(-2x + 1)) - 1| + 1)) - 1 and g(x) = -2((1x - 2)/(-3x - 3)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "How many solutions does the equation -5(1cos(2π((-3(-2|x - 3| - 2) + 3)/(1(-2|x - 3| - 2) + 2)) + 3) - 1)^2 - 4(1cos(2π((-3(-2|x - 3| - 2) + 3)/(1(-2|x - 3| - 2) + 2)) + 3) - 1) + 5 = (-3(1sin(3π(-2|x - 1| + 3) + 2) - 3) - 2)/(-2(1sin(3π(-2|x - 1| + 3) + 2) - 3) + 1) have for x in the interval [-1... | 105 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-1|(2x^2 + 1x - 2) - 2| - 2) - 5 and g(x) = -4(-3cos(1πx + 1)) + 2 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) = -1|(1((2x + 3)/(2x - 3)) - 2) + 3| + 3 and g(x) = -1cos(1π((-2x - 1)/(1x)) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-2((1x + 3)/(-1x - 2))^2 - 1((1x + 3)/(-1x - 2)) - 5) + 1 and g(x) = (-2(-2sin(3πx + 3) + 1) + 3)/(3(-2sin(3πx + 3) + 1) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 57 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(2π(-5(-3x + 2)^2 - 1(-3x + 2) - 4)) + 2 and g(x) = -1((3x - 3)/(1x + 1))^2 + 5((3x - 3)/(1x + 1)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 919 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(1π(3(-5(-2|x - 2| + 2))^2 - 2(-5(-2|x - 2| + 2)) - 5) - 1) and g(x) = (3(-5(3|x - 2| + 3) - 5))/(-3(-5(3|x - 2| + 3) - 5) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 1399 | MATH |
[
{
"content": "How many solutions does the equation 1sin(1π(2((3(1x + 5))/(3(1x + 5) + 1))^2 + 2((3(1x + 5))/(3(1x + 5) + 1)) + 2) - 1) + 3 = -4((1(-3cos(3πx + 1)))/(1(-3cos(3πx + 1)) - 2)) - 4 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|((-2(-2(5x^2 - 2x + 2) + 2) - 3)/(2(-2(5x^2 - 2x + 2) + 2) - 1)) + 3| - 3 and g(x) = 1sin(1π(1((-3x - 2)/(-2x)) - 1) + 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 33 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(1π(2|((-2x - 2)/(1x + 3)) - 2| + 3) + 2) and g(x) = (-1(-5x + 4) + 3)/(-1(-5x + 4) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 84 | MATH |
[
{
"content": "How many solutions does the equation (-3(3cos(2π(-2(-3|x - 1|)^2 + 2(-3|x - 1|) + 1) + 2) - 1) + 3)/(2(3cos(2π(-2(-3|x - 1|)^2 + 2(-3|x - 1|) + 1) + 2) - 1) + 2) = -2cos(2π((-3(1x) - 2)/(-1(1x) - 1)) + 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your... | 586 | MATH |
[
{
"content": "How many solutions does the equation -4(-1(2cos(2πx + 2) + 2) - 5)^2 - 5(-1(2cos(2πx + 2) + 2) - 5) + 3 = -2(-1x + 5)^2 + 2(-1x + 5) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(3π(-5((-3(-2x - 5) + 3)/(3(-2x - 5) + 2))^2 + 2((-3(-2x - 5) + 3)/(3(-2x - 5) + 2)) - 4) - 2) + 1 and g(x) = -1(1cos(2π((-1x)/(3x + 3))))^2 - 4(1cos(2π((-1x)/(3x + 3)))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex form... | 116 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-2cos(2π(1x - 5) - 1) + 1)^2 + 1(-2cos(2π(1x - 5) - 1) + 1) + 3 and g(x) = 2cos(3π((3x - 2)/(-1x + 3))) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 136 | MATH |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.