id stringlengths 34 34 | setting_key stringclasses 1 value | setting_name stringclasses 0 values | messages listlengths 1 1 | ground_truth stringclasses 166 values | dataset stringclasses 1 value |
|---|---|---|---|---|---|
trans_func_intersection_train_0500 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (-2(-5(3|x - 2| - 2)^2 - 4(3|x - 2| - 2) - 1) - 3)/(2(-5(3|x - 2| - 2)^2 - 4(3|x - 2| - 2) - 1)) and g(x) = -2(-2cos(1πx + 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}... | 20 | math |
trans_func_intersection_train_0501 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2|(-1x + 1) + 2| and g(x) = -1x - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0502 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (1(-2sin(2π(-2|x + 3| + 1) + 2) - 3) - 3)/(-1(-2sin(2π(-2|x + 3| + 1) + 2) - 3)) and g(x) = 2(-2x - 2)^2 + 4(-2x - 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
... | 4 | math |
trans_func_intersection_train_0503 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2|(4(-1cos(2π(2x^2 - 2x + 3) + 1) - 3) - 4)| and g(x) = -2|(5((2x)/(-2x + 3))^2 + 5) - 1| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 127 | math |
trans_func_intersection_train_0504 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 5(-1((3(1|x - 2| - 1) - 2)/(3(1|x - 2| - 1) - 1))^2 + 4((3(1|x - 2| - 1) - 2)/(3(1|x - 2| - 1) - 1)) - 4) and g(x) = -2cos(2π(3|((1x - 1)/(-2x - 3)) + 2| + 1) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in La... | 4 | math |
trans_func_intersection_train_0505 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1(-3|x - 3| + 2)^2 - 1 = (2x - 2)/(-1x - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0506 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 3|(1sin(1π(1(-5x + 1)^2 + 5(-5x + 1)) - 3) - 2) - 2| = 1(4((-1x - 2)/(2x))^2 - 2((-1x - 2)/(2x)) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 14 | math |
trans_func_intersection_train_0507 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -2sin(1π((-1(-5(-2x^2 - 2x - 5) + 2) + 3)/(-2(-5(-2x^2 - 2x - 5) + 2) - 1))) - 3 = -1|(-1((2x + 1)/(-1x - 1)) - 2) + 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0508 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 1e^(1((-1(2(1|x| - 2) + 2))/(1(2(1|x| - 2) + 2) + 1))) and g(x) = 3|((1(-2x^2 + 4x - 4))/(-3(-2x^2 + 4x - 4) + 4)) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0509 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 5(-4((1(2|x + 2| - 3) - 2)/(-2(2|x + 2| - 3) - 3)) + 5)^2 - 2(-4((1(2|x + 2| - 3) - 2)/(-2(2|x + 2| - 3) - 3)) + 5) + 3 and g(x) = 2((-3(1|x + 3|) + 1)/(-1(1|x + 3|) + 3))^2 - 3 in the range -10 ≤ x ≤ 10?\n\nPre... | 4 | math |
trans_func_intersection_train_0510 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 4(-2|(-3cos(2π((1x + 3)/(3x + 3)) - 3) + 2) - 3| + 3)^2 - 4(-2|(-3cos(2π((1x + 3)/(3x + 3)) - 3) + 2) - 3| + 3) + 1 and g(x) = -3e^(-2(1|(2x^2 - 2) + 2| + 1)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex f... | 71 | math |
trans_func_intersection_train_0511 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 2sin(3π(1|(1x^2 - 4x - 2)|) - 1) - 2 and g(x) = -3|(5x^2 - 5x + 1) + 2| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0512 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 1(-3sin(3πx - 3)) - 3 = 3|x + 2| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0513 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 3|(-3((3(-4x^2 + 3) + 3)/(1(-4x^2 + 3))) + 3) - 3| and g(x) = (3(2|(4x^2 + 1x - 5) + 2| - 3))/(1(2|(4x^2 + 1x - 5) + 2| - 3) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 7 | math |
trans_func_intersection_train_0514 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (-1(2(-1|x| + 2) + 4) + 1)/(1(2(-1|x| + 2) + 4)) and g(x) = 2|(-2sin(2πx + 3) + 3) + 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0515 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (1(2e^(1(3(1|x| - 2) - 5)) - 2) - 1)/(-2(2e^(1(3(1|x| - 2) - 5)) - 2) + 3) and g(x) = (1(-2(-1x)^2 - 4(-1x) - 2) + 3)/(-3(-2(-1x)^2 - 4(-1x) - 2) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex for... | 2 | math |
trans_func_intersection_train_0516 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -5(-3x^2 + 4x) and g(x) = -1x^2 - 1x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0517 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -1|(-1sin(1π(4(4x^2 - 2x - 3) - 5) - 2) + 3) - 2| - 3 and g(x) = -2cos(3π(2(-4x^2) + 4) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 711 | math |
trans_func_intersection_train_0518 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -3|(-2((2x + 1)/(-1x + 2))^2 - 2((2x + 1)/(-1x + 2)) + 3) + 3| - 1 and g(x) = -1(2|x + 3| + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0519 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 1((3x - 3)/(2x + 2))^2 + 2((3x - 3)/(2x + 2)) - 1 and g(x) = (1x + 3)/(-2x - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0520 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2sin(2π((-3(-1x^2 + 3x - 4) - 2)/(3(-1x^2 + 3x - 4) - 2)) - 3) - 1 and g(x) = 3(-3x^2 + 4x + 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0521 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-2(-2cos(3πx + 2) - 2) + 3)/(2(-2cos(3πx + 2) - 2) + 2) and g(x) = -5x^2 + 4x - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 27 | math |
trans_func_intersection_train_0522 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -1(3sin(3πx + 1)) - 2 and g(x) = -1|x + 2| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 36 | math |
trans_func_intersection_train_0523 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1(-1x - 4)^2 - 5(-1x - 4) + 4 = 5x - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0524 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-1(-5(-2sin(2π(1x^2 + 5x) + 3) + 2) + 3) - 2)/(2(-5(-2sin(2π(1x^2 + 5x) + 3) + 2) + 3) + 1) and g(x) = (1(3|(-3e^(2x) - 1) - 1| - 2) + 2)/(1(3|(-3e^(2x) - 1) - 1| - 2) - 3) where -10 ≤ x ≤ 10.\n\nPresent ... | 99 | math |
trans_func_intersection_train_0525 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -4(-3|x + 3|)^2 - 3(-3|x + 3|) + 3 and g(x) = -3|x + 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0526 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1|(-3x - 2) - 3| + 3 and g(x) = 1x^2 + 5x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0527 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 2|(2sin(3π((-1x + 2)/(-2x - 1)) + 3)) - 2| - 2 and g(x) = 4(2cos(2πx + 3) - 3)^2 + 1(2cos(2πx + 3) - 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 16 | math |
trans_func_intersection_train_0528 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 1|((-2x - 3)/(-1x + 3)) + 1| = -2|x + 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0529 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (2(1(2cos(1π(3|x + 2| - 3) - 2) - 2)^2 + 3(2cos(1π(3|x + 2| - 3) - 2) - 2) + 2) + 2)/(2(1(2cos(1π(3|x + 2| - 3) - 2) - 2)^2 + 3(2cos(1π(3|x + 2| - 3) - 2) - 2) + 2) + 3) and g(x) = 3sin(1π(-5(-2|x - 3| + 2)^2 - 4(-2|x - 3| + 2) - 3)... | 1450 | math |
trans_func_intersection_train_0530 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 3|(-4((1x + 1)/(2x + 1))^2 - 1((1x + 1)/(2x + 1)) - 2) + 2| and g(x) = (2(-3x + 5))/(1(-3x + 5) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0531 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (-3(-3x - 2) + 1)/(-1(-3x - 2) - 2) = 3|x + 3| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0532 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (-2(3(-3|x - 3| + 3) + 4) - 2)/(1(3(-3|x - 3| + 3) + 4) - 2) and g(x) = -3((-1x - 3)/(-2x + 3))^2 + 1((-1x - 3)/(-2x + 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0533 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -1sin(2π(5(-2|((2x + 1)/(-3x + 2)) + 1| + 1)^2 - 4)) + 1 and g(x) = -2|(1((-3x + 3)/(-1x + 2))^2 - 4((-3x + 3)/(-1x + 2)) + 5) - 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Y... | 36 | math |
trans_func_intersection_train_0534 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2|(-3(-2sin(3πx + 2) - 1)^2 - 2(-2sin(3πx + 2) - 1) + 1) + 1| - 1 and g(x) = 1sin(2π(-1x + 2) + 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 185 | math |
trans_func_intersection_train_0535 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 3e^(1(4((-2x)/(-2x + 3))^2 + 4((-2x)/(-2x + 3)) - 3)) + 2 and g(x) = 4((3x + 1)/(1x + 1)) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0536 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -4((-3x + 3)/(-2x + 3))^2 - 1 and g(x) = -3x in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0537 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -2(1cos(1π((-2x + 3)/(1x + 1))) + 3)^2 - 3(1cos(1π((-2x + 3)/(1x + 1))) + 3) - 1 = -2(1x - 5)^2 + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0538 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 4((-2x + 1)/(-1x + 3))^2 - 4((-2x + 1)/(-1x + 3)) - 2 = -1x^2 - 2x - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0539 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -3((2(1|(-3e^(1x) + 1) + 3|))/(1(1|(-3e^(1x) + 1) + 3|) + 2))^2 - 1((2(1|(-3e^(1x) + 1) + 3|))/(1(1|(-3e^(1x) + 1) + 3|) + 2)) = -1(2e^(1(5x^2 + 2x - 3)) - 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \... | 2 | math |
trans_func_intersection_train_0540 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2cos(2π(-1((-2x - 2)/(-1x + 2))^2 - 1((-2x - 2)/(-1x + 2)) + 5) + 3) + 2 and g(x) = -1|(-4x^2 - 3x + 2) + 2| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 15 | math |
trans_func_intersection_train_0541 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 3(-1|((-1(1e^(-2x) + 2) + 2)/(-2(1e^(-2x) + 2) - 2))|)^2 + 4(-1|((-1(1e^(-2x) + 2) + 2)/(-2(1e^(-2x) + 2) - 2))|) and g(x) = 3cos(2π((-3(4x^2 + 2x + 3) - 3)/(2(4x^2 + 2x + 3) - 1)) + 3) - 3 intersect in the interval [-10, 10].\n\nPr... | 3 | math |
trans_func_intersection_train_0542 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -5(1|(5((2x + 1)/(-1x - 2)) + 1) - 3| + 1)^2 + 1(1|(5((2x + 1)/(-1x - 2)) + 1) - 3| + 1) - 3 = 1((2(2sin(3πx - 1) - 3) + 3)/(-1(2sin(3πx - 1) - 3) - 3))^2 + 5((2(2sin(3πx - 1) - 3) + 3)/(-1(2sin(3πx - 1) - 3) - 3)) - 3 have for x in the inte... | 2 | math |
trans_func_intersection_train_0543 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3|(4x)| + 1 and g(x) = -3x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0544 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 4(-3sin(3πx - 3) + 1) + 3 and g(x) = (1x + 1)/(2x) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 61 | math |
trans_func_intersection_train_0545 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1(-2cos(3π((2(2|x + 1|) - 3)/(2(2|x + 1|))) + 1) + 2)^2 - 5(-2cos(3π((2(2|x + 1|) - 3)/(2(2|x + 1|))) + 1) + 2) - 1 and g(x) = (3(-3cos(1π(-3|x - 2|) + 1) + 3) - 3)/(3(-3cos(1π(-3|x - 2|) + 1) + 3) + 1) in the r... | 9 | math |
trans_func_intersection_train_0546 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (1(3|(-1(-3cos(2πx + 2) + 3)^2 + 4(-3cos(2πx + 2) + 3) + 3) - 1| + 1))/(-2(3|(-1(-3cos(2πx + 2) + 3)^2 + 4(-3cos(2πx + 2) + 3) + 3) - 1| + 1) - 2) and g(x) = -1|(3e^(-1((-3x + 1)/(3x + 1))) - 3) - 3| + 3 w... | 4 | math |
trans_func_intersection_train_0547 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -2sin(3π((-2x + 3)/(-1x - 3)) - 1) - 3 and g(x) = (-2x + 2)/(3x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 83 | math |
trans_func_intersection_train_0548 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2|((-2(1(5x^2 + 3x - 4) - 4) - 2)/(-1(1(5x^2 + 3x - 4) - 4) + 3)) + 1| - 3 and g(x) = -1|(-3sin(1π((2x - 1)/(1x - 3)) - 3)) - 2| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your... | 2 | math |
trans_func_intersection_train_0549 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (3(-3cos(1πx + 2) - 3) + 3)/(-2(-3cos(1πx + 2) - 3)) and g(x) = -1x^2 - 2x - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0550 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1(-3|x - 1| - 1) + 4 and g(x) = 1x^2 - 2x + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0551 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -4(3sin(1π(2(3|x - 3| + 2)^2 + 5(3|x - 3| + 2) - 1) + 1) + 3) + 5 and g(x) = -1cos(1π(-3|(-2x^2 + 5x - 1) + 3| - 2) - 3) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 1167 | math |
trans_func_intersection_train_0552 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (3(1(-1cos(3π(3|x - 2| - 1) - 1) - 3) + 4) + 3)/(1(1(-1cos(3π(3|x - 2| - 1) - 1) - 3) + 4) + 3) and g(x) = 4(2((-2x - 2)/(2x)) - 1)^2 + 2(2((-2x - 2)/(2x)) - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex... | 2 | math |
trans_func_intersection_train_0553 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3(1cos(2πx - 3) + 1) - 3 and g(x) = -2x^2 - 1x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0554 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1sin(2π((3x + 1)/(-3x)) - 2) - 3 = -5x^2 + 1x + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0555 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1|(-2((-1(2x + 1) + 2)/(-2(2x + 1) - 1))^2 - 4((-1(2x + 1) + 2)/(-2(2x + 1) - 1)) - 3) + 3| and g(x) = 3|((2(-2sin(1πx + 1) - 2) - 1)/(1(-2sin(1πx + 1) - 2) - 3)) - 1| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent t... | 2 | math |
trans_func_intersection_train_0556 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 3|(2((-1(5x^2 - 5x + 2) - 1)/(-2(5x^2 - 5x + 2) + 1))) - 1| - 1 and g(x) = 3|(1cos(3π(1x + 1) - 3) + 1) + 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0557 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (-2(5(2e^(2(-1x - 4)) - 3)^2 + 5(2e^(2(-1x - 4)) - 3) - 2) - 1)/(-3(5(2e^(2(-1x - 4)) - 3)^2 + 5(2e^(2(-1x - 4)) - 3) - 2)) = (1(-4(1cos(2πx - 1) + 2) + 3) + 3)/(2(-4(1cos(2πx - 1) + 2) + 3) + 1) have for x in the interval [-10, 10]?\n\nPres... | 2 | math |
trans_func_intersection_train_0558 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1(2|((-2x - 2)/(2x - 1)) + 1| - 3) and g(x) = (-1(2sin(3πx) + 1) - 3)/(-2(2sin(3πx) + 1) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 58 | math |
trans_func_intersection_train_0559 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 1((1x + 1)/(-1x - 3))^2 - 3((1x + 1)/(-1x - 3)) + 5 = 3x^2 - 4x - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0560 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1((2(-1|x + 2| + 1) + 1)/(1(-1|x + 2| + 1) - 3))^2 - 1((2(-1|x + 2| + 1) + 1)/(1(-1|x + 2| + 1) - 3)) - 3 and g(x) = -1(4x + 5)^2 + 1(4x + 5) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Y... | 2 | math |
trans_func_intersection_train_0561 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (-1(1|(3e^(2x) - 1)| - 2) - 3)/(3(1|(3e^(2x) - 1)| - 2) + 2) = 1(-1x^2 + 5x + 3) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0562 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -4((-3(3|x - 2| - 3) + 1)/(-2(3|x - 2| - 3) + 3))^2 + 2((-3(3|x - 2| - 3) + 1)/(-2(3|x - 2| - 3) + 3)) - 1 and g(x) = -5(1e^(1x))^2 + 3(1e^(1x)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex form... | 3 | math |
trans_func_intersection_train_0563 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -1sin(1π(-2|x + 1| + 2) - 3) and g(x) = 2x^2 - 2x - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0564 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 3sin(2π((-1(1|(5x - 1) - 2| - 3) + 1)/(3(1|(5x - 1) - 2| - 3) - 2)) - 3) - 2 and g(x) = (-3(-2|(1x - 5) + 3| + 2) + 3)/(1(-2|(1x - 5) + 3| + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed... | 18 | math |
trans_func_intersection_train_0565 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-1(-5(-2|x + 2| - 2) + 2) - 1)/(2(-5(-2|x + 2| - 2) + 2)) and g(x) = (-2(-2cos(3πx + 3) + 1) + 2)/(-3(-2cos(3πx + 3) + 1)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 60 | math |
trans_func_intersection_train_0566 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 3|(-2cos(2π(2x) + 1) - 3) + 2| + 2 and g(x) = 5(3x)^2 - 2(3x) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0567 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 1|(5(2e^(1((-3x - 2)/(1x + 1))) + 2)) - 3| + 2 and g(x) = 2e^(2(-5(3|x - 3| - 2) - 4)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0568 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 3cos(1π((3(-1x + 5) + 3)/(3(-1x + 5) - 2)) - 2) - 3 = -3sin(2π((2x - 2)/(-3x + 2))) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 58 | math |
trans_func_intersection_train_0569 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -4((2(-5x + 2) + 1)/(-1(-5x + 2) - 1))^2 + 5((2(-5x + 2) + 1)/(-1(-5x + 2) - 1)) - 2 = -3((-2x + 1)/(-1x + 3)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0570 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -1|((3(-3(-1sin(1πx)) - 2) - 2)/(1(-3(-1sin(1πx)) - 2) + 3)) + 2| + 2 and g(x) = 2(1(-3sin(3πx - 2) + 1) - 3)^2 - 5(1(-3sin(3πx - 2) + 1) - 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format... | 40 | math |
trans_func_intersection_train_0571 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-3(2(-3(-1|x + 3|) - 1)^2 - 1(-3(-1|x + 3|) - 1) - 4) - 1)/(3(2(-3(-1|x + 3|) - 1)^2 - 1(-3(-1|x + 3|) - 1) - 4) - 2) and g(x) = -4((1(1x + 3) + 2)/(2(1x + 3)))^2 - 4((1(1x + 3) + 2)/(2(1x + 3))) - 1 wher... | 4 | math |
trans_func_intersection_train_0572 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 5(-1(-2|((-2x + 2)/(-2x)) - 2| + 2)^2 + 2(-2|((-2x + 2)/(-2x)) - 2| + 2) - 2) + 5 and g(x) = -1cos(2π(2((-3x - 1)/(-3x - 3)) + 5)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Yo... | 22 | math |
trans_func_intersection_train_0573 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1|(-5(2((3x)/(-1x + 2))^2) + 1) + 3| - 1 and g(x) = -4((1(-2x + 4) - 1)/(3(-2x + 4) + 2))^2 + 1((1(-2x + 4) - 1)/(3(-2x + 4) + 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 2 | math |
trans_func_intersection_train_0574 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -2((2x - 3)/(3x + 1))^2 - 1((2x - 3)/(3x + 1)) - 5 = -2|x + 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0575 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-3(5(-4x - 1)^2 + 2(-4x - 1)) - 2)/(2(5(-4x - 1)^2 + 2(-4x - 1)) - 1) and g(x) = 4(-1sin(3πx)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 62 | math |
trans_func_intersection_train_0576 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 2sin(3π(-3(-5x^2 + 2x + 5) - 1) + 2) - 2 and g(x) = -1e^(-2(-5x)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 768 | math |
trans_func_intersection_train_0577 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -4((-1(1|(-2sin(1πx)) + 2| - 2) + 1)/(1(1|(-2sin(1πx)) + 2| - 2) + 1)) and g(x) = (-2(-1sin(1π(-2x - 2) - 1) + 2))/(1(-1sin(1π(-2x - 2) - 1) + 2)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format... | 20 | math |
trans_func_intersection_train_0578 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2(5x + 5)^2 - 3(5x + 5) and g(x) = 3|x + 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0579 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (1(-1sin(2πx - 1) + 3))/(3(-1sin(2πx - 1) + 3) + 2) = -3|x + 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0580 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -5(-2cos(1πx + 1) - 2)^2 + 5(-2cos(1πx + 1) - 2) + 1 and g(x) = -2x + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 8 | math |
trans_func_intersection_train_0581 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 3(2|x - 3| + 3) - 1 and g(x) = 3|x + 1| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0582 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2|(-4((-3(5x^2 - 2x + 4) - 3)/(3(5x^2 - 2x + 4))) - 4) + 2| - 3 and g(x) = (1(-1(3sin(2πx - 3) + 3) + 3) - 3)/(-1(-1(3sin(2πx - 3) + 3) + 3) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: ... | 40 | math |
trans_func_intersection_train_0583 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (1(2|(-1(-4x^2 + 1) - 4) - 3| + 3) + 2)/(1(2|(-1(-4x^2 + 1) - 4) - 3| + 3) - 3) and g(x) = 1|((-3(3e^(1x) - 3) + 3)/(-3(3e^(1x) - 3) - 2)) + 2| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex forma... | 2 | math |
trans_func_intersection_train_0584 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2(1cos(2π(1|x| + 1) + 1) - 3)^2 - 3(1cos(2π(1|x| + 1) + 1) - 3) + 4 and g(x) = 3|(-1x^2 + 1x) - 2| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
trans_func_intersection_train_0585 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-2(2e^(-2x)) + 2)/(-3(2e^(-2x))) and g(x) = (-2x - 1)/(3x) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0586 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -5(2sin(2πx + 3) + 2) - 3 and g(x) = 1x + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 10 | math |
trans_func_intersection_train_0587 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 4(2sin(3π((-1(-1x^2 - 2x - 5) - 3)/(-2(-1x^2 - 2x - 5) - 3))) + 3) - 1 and g(x) = (1(-2(-1x - 1)^2 + 4(-1x - 1) + 5) + 1)/(1(-2(-1x - 1)^2 + 4(-1x - 1) + 5) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex fo... | 2 | math |
trans_func_intersection_train_0588 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -1(-2((1x - 1)/(1x + 3)) - 5)^2 + 4(-2((1x - 1)/(1x + 3)) - 5) + 2 and g(x) = (1(1e^(-1x) + 2) - 3)/(1(1e^(-1x) + 2) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
... | 2 | math |
trans_func_intersection_train_0589 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2cos(3π((2x + 1)/(-3x))) - 1 and g(x) = -2x^2 + 2x + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0590 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -1((-2(-1(-2sin(1πx - 2) + 2) + 4))/(-1(-1(-2sin(1πx - 2) + 2) + 4)))^2 - 1((-2(-1(-2sin(1πx - 2) + 2) + 4))/(-1(-1(-2sin(1πx - 2) + 2) + 4))) and g(x) = -2((-2(2cos(3πx - 3) - 2) - 1)/(-1(2cos(3πx - 3) - ... | 60 | math |
trans_func_intersection_train_0591 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2(-1cos(2πx - 1) - 3) - 3 and g(x) = 1x - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
trans_func_intersection_train_0592 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3cos(3π((1(-3x - 3) + 3)/(-2(-3x - 3) - 2)) - 2) + 1 and g(x) = (2(-2|x - 1| - 1) + 3)/(3(-2|x - 1| - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
trans_func_intersection_train_0593 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -2((2(-1|(1x^2 + 5x + 4) - 3| - 2) + 1)/(1(-1|(1x^2 + 5x + 4) - 3| - 2) + 3)) = (1(5(-3|x - 2| + 2)^2 - 4(-3|x - 2| + 2) - 5) - 3)/(1(5(-3|x - 2| + 2)^2 - 4(-3|x - 2| + 2) - 5)) have for x in the interval [-10, 10]?\n\nPresent the answer in ... | 4 | math |
trans_func_intersection_train_0594 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 2(1(1|x + 3| - 1)^2 + 1(1|x + 3| - 1) - 3) - 5 and g(x) = -1cos(1π(3|x - 2|) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0595 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-1(5(2x - 2)^2 + 1(2x - 2)))/(2(5(2x - 2)^2 + 1(2x - 2)) - 3) and g(x) = -1sin(3π(-2x^2 - 2x - 1) + 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0596 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -4((1(-1sin(3π(1|x + 1| + 3) - 1) - 2) - 3)/(-2(-1sin(3π(1|x + 1| + 3) - 1) - 2)))^2 + 4((1(-1sin(3π(1|x + 1| + 3) - 1) - 2) - 3)/(-2(-1sin(3π(1|x + 1| + 3) - 1) - 2))) - 1 and g(x) = -2((2(-2|x + 3| + 2) + 1)/(2(-2|x + 3| + 2) - 1)... | 4 | math |
trans_func_intersection_train_0597 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 5((-1(-1(-1|x| + 3) - 4) - 3)/(3(-1(-1|x| + 3) - 4) + 3))^2 - 1 and g(x) = 1cos(1π((-1(5x + 5) + 1)/(-2(5x + 5) + 3)) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
... | 8 | math |
trans_func_intersection_train_0598 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -4((2(2e^(2x) - 2) - 2)/(-3(2e^(2x) - 2) + 2))^2 + 4 and g(x) = (2(3cos(2πx + 2) + 2) - 2)/(-3(3cos(2πx + 2) + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 42 | math |
trans_func_intersection_train_0599 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2(3cos(2πx + 3) + 3) + 2 and g(x) = -1x^2 - 1x + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 8 | math |
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