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_0400 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1(3(-3sin(1πx - 3) - 1)^2 - 5(-3sin(1πx - 3) - 1) - 3) = 5((-3x + 2)/(1x + 3)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
trans_func_intersection_train_0401 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 3cos(2π(-3x + 1) - 3) + 1 and g(x) = -2|x - 1| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 30 | math |
trans_func_intersection_train_0402 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1(1|(-4x) + 1| - 1)^2 + 3(1|(-4x) + 1| - 1) + 5 = -1|(2cos(1πx + 3) - 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_0403 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1((2(-2sin(3πx - 2) - 1))/(1(-2sin(3πx - 2) - 1) + 2))^2 - 4((2(-2sin(3πx - 2) - 1))/(1(-2sin(3πx - 2) - 1) + 2)) + 1 and g(x) = -4((-2x + 1)/(2x + 1))^2 + 2((-2x + 1)/(2x + 1)) + 1 intersect in the interval [-10, 10].\n\nPresent t... | 70 | math |
trans_func_intersection_train_0404 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2|(-4(2((-1x)/(-1x + 2)) + 5)^2)| + 1 and g(x) = -1sin(3π(-5(1|x + 1|)^2 + 3(1|x + 1|) + 2) - 3) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 5 | math |
trans_func_intersection_train_0405 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2|(1((3(1x^2 + 5x + 3))/(-2(1x^2 + 5x + 3) - 2)) - 1) - 1| + 3 and g(x) = (-3(-4(-1e^(2x) + 1) + 4))/(-1(-4(-1e^(2x) + 1) + 4) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your... | 4 | math |
trans_func_intersection_train_0406 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3|((3(-5(-2x^2 + 2x + 1) + 4))/(1(-5(-2x^2 + 2x + 1) + 4) - 3)) + 3| + 1 and g(x) = -2|(2(-3e^(1x) + 2) - 3) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 5 | math |
trans_func_intersection_train_0407 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -2e^(-1((-1(-2|(-1x^2 + 1x + 2)|) - 1)/(2(-2|(-1x^2 + 1x + 2)|) - 1))) + 2 = 1|(-1((1x + 1)/(3x)) + 2) - 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
trans_func_intersection_train_0408 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 1(-2x + 1)^2 + 1(-2x + 1) - 5 and g(x) = 2|x + 3| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0409 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2|(-2e^(-2x) - 2) + 2| - 3 and g(x) = -3|x - 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0410 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 2((-3(3|(3sin(2πx) + 3) - 2| + 2) + 3)/(-2(3|(3sin(2πx) + 3) - 2| + 2)))^2 - 4 and g(x) = -5(2(1e^(1x))^2 - 5(1e^(1x)) + 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0411 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 2(1x - 4)^2 + 3(1x - 4) + 1 and g(x) = 2x - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0412 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -1(-3|x + 3| + 1)^2 - 5(-3|x + 3| + 1) - 4 and g(x) = 3x^2 + 2x - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0413 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3(-1((-2x + 3)/(-1x + 2))^2 - 5((-2x + 3)/(-1x + 2)) + 4) + 5 and g(x) = 2(-3|x + 3| - 1)^2 + 2(-3|x + 3| - 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0414 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1|(3((1(3x))/(-1(3x) + 2))^2 - 1((1(3x))/(-1(3x) + 2)) + 3) + 2| + 1 = -3|(2((1x)/(2x + 2)) - 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_0415 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 3((1(4(1|x + 3| + 3)^2) - 2)/(-1(4(1|x + 3| + 3)^2) - 3)) + 5 and g(x) = -3cos(1π(-1|(5x + 3) + 3|) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 100 | math |
trans_func_intersection_train_0416 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -4(-2sin(2πx) - 3) - 3 and g(x) = 4x^2 + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 8 | math |
trans_func_intersection_train_0417 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2sin(1π(1x^2 - 3x - 1) - 3) + 3 and g(x) = 3|x + 2| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 20 | math |
trans_func_intersection_train_0418 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 4(1x^2 + 5x + 3) + 4 and g(x) = -1x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0419 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-2(2|(-5x - 2) + 3|) + 2)/(3(2|(-5x - 2) + 3|) - 1) and g(x) = -2|(2x + 4) + 2| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0420 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-2(2e^(2x) + 1) + 3)/(2(2e^(2x) + 1) + 2) and g(x) = -1|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_0421 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (-1(-3|(1cos(2πx) + 2) - 3| - 2) + 3)/(3(-3|(1cos(2πx) + 2) - 3| - 2) + 2) = 1(1x^2 - 5x + 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_0422 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (-2(-4(2|(3x^2 + 3x - 3) - 2| - 1) + 4) + 3)/(-2(-4(2|(3x^2 + 3x - 3) - 2| - 1) + 4) - 1) = -1(-2cos(2π(-2|x - 2| - 1) + 1) - 1)^2 - 1(-2cos(2π(-2|x - 2| - 1) + 1) - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex... | 158 | math |
trans_func_intersection_train_0423 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-3(-1|(2(4x^2 - 5x - 3) + 1) - 3| + 3) - 3)/(-2(-1|(2(4x^2 - 5x - 3) + 1) - 3| + 3) - 1) and g(x) = -1|((1(-1x^2 + 4x - 3) - 1)/(-3(-1x^2 + 4x - 3)))| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaT... | 4 | math |
trans_func_intersection_train_0424 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2|(-1((1x + 1)/(1x)) - 1) - 2| - 3 and g(x) = (-3(-1x^2 - 4x + 2) - 2)/(3(-1x^2 - 4x + 2) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0425 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 2(1|(-2((-2x + 3)/(-1x + 2))) - 3| - 1)^2 - 4(1|(-2((-2x + 3)/(-1x + 2))) - 3| - 1) - 5 = 5(-1e^(1((3x)/(-3x + 1))) + 2)^2 + 1(-1e^(1((3x)/(-3x + 1))) + 2) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 2 | math |
trans_func_intersection_train_0426 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-3(1|(1(-3x - 1)^2 + 1(-3x - 1) - 5) - 1| - 1) - 3)/(3(1|(1(-3x - 1)^2 + 1(-3x - 1) - 5) - 1| - 1) - 2) and g(x) = 3sin(1π((-2(4x^2 + 2x + 1) - 3)/(-2(4x^2 + 2x + 1) - 3)) - 1) + 2 where -10 ≤ x ≤ 10.\n\n... | 4 | math |
trans_func_intersection_train_0427 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (-1(1|(-3x^2 + 5x - 1) + 2| + 2) - 3)/(-3(1|(-3x^2 + 5x - 1) + 2| + 2) - 3) = 3cos(3π(4x^2 + 3x + 1) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1478 | math |
trans_func_intersection_train_0428 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -5(3x^2 - 3x - 1) - 2 and g(x) = -1|x + 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0429 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1|(-5(3((-2x + 1)/(3x))^2 - 1) - 5) + 3| and g(x) = 2|((1(3cos(1πx - 2) + 2) + 2)/(2(3cos(1πx - 2) + 2))) - 1| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 38 | math |
trans_func_intersection_train_0430 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -2(-2|(1((3x - 2)/(2x))^2 + 4((3x - 2)/(2x))) + 1| + 2) + 3 and g(x) = -4(-3cos(1π(-4x^2 + 2x + 3) - 2) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0431 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (-2(-1(-3cos(1π(-2|x| + 2) + 3) - 1)^2 + 5(-3cos(1π(-2|x| + 2) + 3) - 1) + 3) - 2)/(-2(-1(-3cos(1π(-2|x| + 2) + 3) - 1)^2 + 5(-3cos(1π(-2|x| + 2) + 3) - 1) + 3) - 1) = 2|(-3(-2cos(1πx + 1) + 2)^2 + 3(-2cos(1πx + 1) + 2) + 2) + 2| + 2 have fo... | 9 | math |
trans_func_intersection_train_0432 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -2cos(1π((3(3(1x^2 + 1x) - 3) - 2)/(3(3(1x^2 + 1x) - 3) - 2)) + 3) - 2 = -2|(-1((-2x - 3)/(1x))) + 3| - 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_0433 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 5(3cos(2πx - 2) + 1)^2 - 2(3cos(2πx - 2) + 1) - 3 and g(x) = -1x - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 45 | math |
trans_func_intersection_train_0434 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 2(2sin(3π((-3x - 2)/(1x + 2))) + 2) + 2 = 3|(4x^2 + 1x + 3) - 2| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
trans_func_intersection_train_0435 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -1|((3x - 1)/(2x - 1)) + 3| + 2 and g(x) = 1x - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0436 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 4(2cos(2π(-3|x - 3| - 1) - 1) - 2)^2 + 2(2cos(2π(-3|x - 3| - 1) - 1) - 2) - 3 and g(x) = (2(1x^2 + 3x + 5) - 2)/(1(1x^2 + 3x + 5)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your a... | 120 | math |
trans_func_intersection_train_0437 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1sin(1π(1|x + 2| + 1) - 2) + 3 and g(x) = 1x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0438 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 4((-2(-1x^2 + 4) - 1)/(3(-1x^2 + 4) - 2)) - 2 and g(x) = 1|(2sin(3πx - 1) + 2) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0439 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2|((2(-2(-3x + 3)^2 + 2(-3x + 3) + 4) + 3)/(-3(-2(-3x + 3)^2 + 2(-3x + 3) + 4) + 2)) + 2| - 1 and g(x) = (-3(3(4x + 4)^2 + 5(4x + 4) + 3) + 2)/(1(3(4x + 4)^2 + 5(4x + 4) + 3)) in the range -10 ≤ x ≤ 10?\n\nPres... | 2 | math |
trans_func_intersection_train_0440 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2|(3cos(2πx + 2) + 3) - 2| + 2 and g(x) = -5x + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 5 | math |
trans_func_intersection_train_0441 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3cos(2π((-2x)/(-1x + 3)) - 2) + 1 and g(x) = 5x^2 + 4x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0442 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 3|(-4x^2 - 3x + 4)| - 3 = (2x + 1)/(2x - 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_0443 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -4(1(1|((2x - 1)/(1x)) + 2| - 1)^2 + 2(1|((2x - 1)/(1x)) + 2| - 1) - 2) - 2 and g(x) = -1(1((1x + 1)/(-2x - 1)) + 5)^2 - 4(1((1x + 1)/(-2x - 1)) + 5) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\... | 4 | math |
trans_func_intersection_train_0444 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -3(3e^(1x) - 2) - 1 and g(x) = 5x^2 - 1x - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0445 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (-3(2|(4(1e^(-2x) - 1)^2 + 1(1e^(-2x) - 1) - 3) + 2| + 3) + 2)/(-3(2|(4(1e^(-2x) - 1)^2 + 1(1e^(-2x) - 1) - 3) + 2| + 3) - 2) and g(x) = (-2(-1|(1x^2 - 2x - 4) + 3| + 2))/(-1(-1|(1x^2 - 2x - 4) + 3| + 2) + 2) intersect in the interv... | 2 | math |
trans_func_intersection_train_0446 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (1(-3|x| - 2) + 1)/(-2(-3|x| - 2) + 1) = -1|x - 2| + 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_0447 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -3|(3(-3x^2 + 1x + 4) + 4) + 2| + 3 and g(x) = -1cos(2π(-3x^2 + 3x - 2) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0448 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2|((-1x - 3)/(-1x)) + 2| and g(x) = -1x + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0449 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 3|(1x^2) - 2| + 3 and g(x) = 4x^2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0450 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -1sin(3π(1|(1x^2 + 5x - 1)| - 1) - 3) + 3 and g(x) = 2cos(2π(-2x + 5) + 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 214 | math |
trans_func_intersection_train_0451 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 2|(-2sin(3πx + 1) - 1) - 1| + 3 and g(x) = 3|x + 2| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 16 | math |
trans_func_intersection_train_0452 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -4(-2cos(1π(3|x - 3| - 1)) - 3)^2 - 4(-2cos(1π(3|x - 3| - 1)) - 3) and g(x) = 2(-2e^(-1x) - 1) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 39 | math |
trans_func_intersection_train_0453 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 2sin(2π(1(1x - 2)^2 + 5(1x - 2) + 1) + 1) - 3 = 1|((2x + 1)/(2x)) - 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 187 | math |
trans_func_intersection_train_0454 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (-3(-4(-4x^2 - 2x + 5) + 3) - 2)/(-1(-4(-4x^2 - 2x + 5) + 3) + 3) and g(x) = -4(-5x^2 + 1x - 4) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0455 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-3(2e^(2(-2x^2 - 4x - 5)) + 2) + 3)/(-1(2e^(2(-2x^2 - 4x - 5)) + 2) - 3) and g(x) = -2cos(2π((3x + 3)/(2x + 3)) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 45 | math |
trans_func_intersection_train_0456 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (3(-1|(5(1x - 1)^2 - 5(1x - 1) + 2) + 2| - 3) + 2)/(-2(-1|(5(1x - 1)^2 - 5(1x - 1) + 2) + 2| - 3) + 1) and g(x) = -1|(2(2x - 4)^2 + 5(2x - 4)) - 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\bo... | 4 | math |
trans_func_intersection_train_0457 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 1((-2(5x + 3) + 3)/(1(5x + 3)))^2 + 5((-2(5x + 3) + 3)/(1(5x + 3))) + 5 and g(x) = 5(-1|x| + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0458 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -2(2sin(1πx) - 1)^2 + 2(2sin(1πx) - 1) - 5 and g(x) = -3|x + 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 12 | math |
trans_func_intersection_train_0459 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 2cos(2π(-3x^2 - 5x - 2) + 3) - 2 and g(x) = -1|x - 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 419 | math |
trans_func_intersection_train_0460 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2sin(2π((2x)/(2x))) - 2 and g(x) = 1|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_0461 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2sin(3π((-1x - 1)/(2x - 3))) and g(x) = -5x^2 - 5x + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0462 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 1(1|(-3e^(-1x) - 1) - 1| + 1) + 1 and g(x) = (-3(-3cos(1πx + 3) + 3))/(3(-3cos(1πx + 3) + 3) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 13 | math |
trans_func_intersection_train_0463 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 5(3cos(2π(5((2x + 2)/(3x + 3))^2 + 4((2x + 2)/(3x + 3))) - 3)) + 5 and g(x) = 1(-3|(1x^2 + 4x - 4) - 3| + 1) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0464 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -3cos(3π(3(-4((1x - 1)/(2x - 1))^2 + 1((1x - 1)/(2x - 1)) - 2) + 3) + 3) + 1 and g(x) = 2(-4((1x + 3)/(1x - 2))^2 - 5((1x + 3)/(1x - 2)) - 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: ... | 16 | math |
trans_func_intersection_train_0465 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -2cos(3π((-3x + 3)/(-2x - 3)) - 1) and g(x) = 1|x - 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 112 | math |
trans_func_intersection_train_0466 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (1(1(-1|x|)^2 - 5(-1|x|) - 2) - 2)/(3(1(-1|x|)^2 - 5(-1|x|) - 2) + 3) and g(x) = -2|(-1x - 3) - 1| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0467 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -5(-5x^2 - 2x + 5) + 2 and g(x) = -3x + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0468 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-1(1|(3(-3x - 3)^2 - 5(-3x - 3) + 1)|) + 2)/(-2(1|(3(-3x - 3)^2 - 5(-3x - 3) + 1)|) + 3) and g(x) = -4(-4((-3x + 1)/(-1x - 3))^2 - 4((-3x + 1)/(-1x - 3)) + 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answe... | 2 | math |
trans_func_intersection_train_0469 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 2(-3cos(2π(-5x^2 - 3x + 3) - 1)) - 2 and g(x) = 3|(-1x - 5) + 2| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 228 | math |
trans_func_intersection_train_0470 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 3(-1|((-2(-3x))/(-2(-3x) + 3)) + 3| + 3)^2 - 2(-1|((-2(-3x))/(-2(-3x) + 3)) + 3| + 3) - 1 and g(x) = 1sin(3π(-1(1|x - 3| + 1) + 2) + 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\box... | 6 | math |
trans_func_intersection_train_0471 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 4(3sin(2πx) + 1)^2 - 4(3sin(2πx) + 1) + 1 = (2x + 1)/(2x - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 74 | math |
trans_func_intersection_train_0472 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -2|(3cos(1π(-2((-1x + 2)/(2x + 2)) - 4) + 2) + 2) - 1| and g(x) = -2(-2cos(2π(3|x + 3| - 3) + 2) + 1)^2 + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 124 | math |
trans_func_intersection_train_0473 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (2(1|(2x^2 + 3x - 5)| + 2))/(-1(1|(2x^2 + 3x - 5)| + 2) + 3) and g(x) = 3|(5x^2 - 2x + 1) - 3| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0474 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 2e^(2(-2|x + 1| + 1)) - 2 = (-1x - 2)/(-1x + 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_0475 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1|(1((-2x + 3)/(-2x - 1)) - 1) + 2| and g(x) = -2(1|x + 3| + 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0476 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (2(3x^2 - 2x + 4) + 3)/(-1(3x^2 - 2x + 4) - 2) and g(x) = -1x^2 + 3x + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0477 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-2(-2cos(1π(2|(4x^2 - 4x + 3) + 2| - 3) + 2) - 3) - 1)/(-2(-2cos(1π(2|(4x^2 - 4x + 3) + 2| - 3) + 2) - 3) + 1) and g(x) = -3(3|(4x^2 - 1x - 5) + 3| - 2) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in L... | 4 | math |
trans_func_intersection_train_0478 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 1(-2sin(1π((3(-1x^2 + 5x + 5) - 1)/(1(-1x^2 + 5x + 5) - 2)) - 2) + 3) and g(x) = (-1(3|(-4x) + 1|) - 1)/(3(3|(-4x) + 1|) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0479 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (-2(1(-3cos(3πx - 2))^2 - 1(-3cos(3πx - 2)) + 4))/(-2(1(-3cos(3πx - 2))^2 - 1(-3cos(3πx - 2)) + 4) - 1) and g(x) = -4(-3x^2 - 4x + 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 2 | math |
trans_func_intersection_train_0480 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2((-3(-4x - 2) + 1)/(2(-4x - 2)))^2 - 3((-3(-4x - 2) + 1)/(2(-4x - 2))) + 1 and g(x) = -2(1|x + 2| + 1)^2 + 5(1|x + 2| + 1) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 2 | math |
trans_func_intersection_train_0481 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3|((-2(3x^2 + 2x - 3) - 2)/(-1(3x^2 + 2x - 3))) + 1| + 3 and g(x) = 2(-3x - 3)^2 + 4(-3x - 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0482 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 2(-2sin(1π(1x^2 + 1x + 3) + 1) - 2) - 4 = 1(-5x^2 + 5x + 5) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0483 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 2|(-2sin(1πx + 2) + 3) - 3| - 2 = -1x^2 - 4x + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
trans_func_intersection_train_0484 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2((-2(-1e^(1x) + 2) - 3)/(-1(-1e^(1x) + 2) - 1))^2 + 4((-2(-1e^(1x) + 2) - 3)/(-1(-1e^(1x) + 2) - 1)) + 4 and g(x) = 2(-1cos(1πx - 1) + 3) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\b... | 2 | math |
trans_func_intersection_train_0485 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1|(-2x - 3) - 3| + 3 and g(x) = 3|x + 1| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0486 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3|(1x) + 1| + 2 and g(x) = (1x - 1)/(3x - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0487 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1(3x + 3)^2 - 5(3x + 3) - 4 = (-1x - 1)/(-3x - 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_0488 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 1(-1|(-2cos(3π(1x + 5)) + 3) - 1| + 2)^2 - 4(-1|(-2cos(3π(1x + 5)) + 3) - 1| + 2) - 5 and g(x) = (-3(-2|(-5x - 3) - 3| - 1) - 1)/(-3(-2|(-5x - 3) - 3| - 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format:... | 60 | math |
trans_func_intersection_train_0489 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-2(-2sin(2π(1|x| - 3) - 2)) - 1)/(-3(-2sin(2π(1|x| - 3) - 2)) + 1) and g(x) = 3e^(2((2x - 2)/(3x + 1))) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 36 | math |
trans_func_intersection_train_0490 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 5(-2(-3sin(3πx - 1) - 2)^2 - 4(-3sin(3πx - 1) - 2) + 4) + 2 and g(x) = -1|(3x + 2) + 1| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 82 | math |
trans_func_intersection_train_0491 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (1(-2cos(2πx + 2)) - 3)/(-2(-2cos(2πx + 2)) - 2) and g(x) = -3x + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 39 | math |
trans_func_intersection_train_0492 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1|(-3((-1(-4x^2 - 4x + 4) - 2)/(-2(-4x^2 - 4x + 4) + 3)) + 3)| + 3 and g(x) = -1e^(-1((1(1|x + 3| + 1) - 1)/(-2(1|x + 3| + 1) + 1))) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
... | 4 | math |
trans_func_intersection_train_0493 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 5(-3e^(-1(2|x - 1| - 3)) + 2)^2 + 4(-3e^(-1(2|x - 1| - 3)) + 2) + 1 = 4(-2e^(1x) + 3) - 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_0494 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1cos(3π(2|((-2x + 1)/(1x + 1))| - 2)) - 2 and g(x) = -1cos(1π(5x^2 + 3x + 4) - 1) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 671 | math |
trans_func_intersection_train_0495 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (3(-5x^2 + 3x + 5))/(-3(-5x^2 + 3x + 5) - 2) and g(x) = (-1x)/(-1x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0496 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 2(-3sin(3π(-1|x - 1| - 3)) - 1)^2 - 5(-3sin(3π(-1|x - 1| - 3)) - 1) + 2 and g(x) = 2|((-2x + 2)/(-1x - 2))| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 56 | math |
trans_func_intersection_train_0497 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -3e^(-1(-5(-2|x| - 3) - 3)) and g(x) = -1(2cos(3πx - 1)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 60 | math |
trans_func_intersection_train_0498 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -5(-3|((3x + 1)/(-3x)) - 3| - 1)^2 + 5(-3|((3x + 1)/(-3x)) - 3| - 1) and g(x) = -3(3|x| - 2)^2 + 3(3|x| - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0499 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (2(2|(-3e^(2x) - 3) - 2| + 1) + 3)/(-3(2|(-3e^(2x) - 3) - 2| + 1) + 2) and g(x) = -2((1x + 2)/(1x - 3))^2 - 4((1x + 2)/(1x - 3)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 2 | math |
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