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_0700 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -3|(2(1((-1x)/(-1x + 2))^2 + 2((-1x)/(-1x + 2)) + 1) - 2) + 1| = -2((-3(-5x^2 + 3x - 2) + 2)/(2(-5x^2 + 3x - 2))) - 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_0701 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 3((3(-3|x - 1| + 3) + 3)/(3(-3|x - 1| + 3) + 1)) - 2 and g(x) = -3(1cos(3πx + 2)) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 58 | math |
trans_func_intersection_train_0702 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3(4(1cos(1π((1x - 1)/(-3x + 2)) - 1) + 2)^2 + 3(1cos(1π((1x - 1)/(-3x + 2)) - 1) + 2) + 2) - 5 and g(x) = -1(-2|((2x)/(3x + 2)) - 1| - 2)^2 - 2(-2|((2x)/(3x + 2)) - 1| - 2) + 5 intersect in the interval [-10, 10].\n\nPresent the an... | 2 | math |
trans_func_intersection_train_0703 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 4((1(-3sin(1πx - 2) + 1) - 1)/(-2(-3sin(1πx - 2) + 1)))^2 + 1((1(-3sin(1πx - 2) + 1) - 1)/(-2(-3sin(1πx - 2) + 1))) + 5 and g(x) = (-1(5x^2 - 5x - 5) + 2)/(-2(5x^2 - 5x - 5) - 1) where -10 ≤ x ≤ 10.\n\nPre... | 2 | math |
trans_func_intersection_train_0704 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (-3(2(-1sin(1πx + 3) - 1) + 2) - 1)/(2(2(-1sin(1πx + 3) - 1) + 2) - 1) = (-1(-1e^(-1x) - 3))/(3(-1e^(-1x) - 3) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 20 | math |
trans_func_intersection_train_0705 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1|(-3((1x - 1)/(-2x + 1))^2 - 2((1x - 1)/(-2x + 1)) - 4) + 2| + 1 = -1(3|x| - 1)^2 + 5(3|x| - 1) + 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_0706 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 3((-3(-1|(-4x^2 + 2x - 2)| - 2) + 2)/(2(-1|(-4x^2 + 2x - 2)| - 2) + 1)) + 1 and g(x) = -3cos(3π(2|(2x + 1) - 2| - 3)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
... | 234 | math |
trans_func_intersection_train_0707 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -4(-3|((1(-1x + 2) + 1)/(3(-1x + 2) + 1)) + 1| - 3)^2 + 1(-3|((1(-1x + 2) + 1)/(3(-1x + 2) + 1)) + 1| - 3) + 3 and g(x) = -4(3|(-3e^(-2x) + 3) + 2| + 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format:... | 3 | math |
trans_func_intersection_train_0708 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -3cos(1π(3x + 3) - 3) + 1 = 3|x + 3| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0709 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2|(-1sin(2πx) - 2) - 2| - 1 and g(x) = 4x + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0710 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1(-3((3x - 3)/(-3x + 1))^2 - 4((3x - 3)/(-3x + 1)) - 4) - 5 and g(x) = -1|((-2x - 2)/(2x + 1)) + 3| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0711 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (2(2e^(2x) - 3))/(1(2e^(2x) - 3)) and g(x) = 1x^2 + 3x - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0712 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2|(-4((-2x + 3)/(2x))^2 - 5((-2x + 3)/(2x)) + 4) + 1| + 1 and g(x) = 3((-1x + 1)/(3x + 1))^2 + 2((-1x + 1)/(3x + 1)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0713 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (2(1(-3x^2 + 2x + 4) - 1) + 1)/(-2(1(-3x^2 + 2x + 4) - 1) + 3) and g(x) = 4(2sin(1πx + 2) - 3) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0714 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 1|(-1e^(-1(4(1x - 1)^2 + 3(1x - 1) + 1)) + 3) + 2| and g(x) = 1((-3(3x^2 + 1x - 3) + 1)/(1(3x^2 + 1x - 3) + 3)) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0715 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3|((1x - 1)/(3x + 3)) - 3| + 3 and g(x) = 4x^2 - 5x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0716 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -3(-2sin(2π(3x^2 + 1x)) + 1) + 2 and g(x) = 2sin(3π((1x + 3)/(1x + 3)) - 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1139 | math |
trans_func_intersection_train_0717 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 2sin(2π(-1|x| + 1) + 1) = (-1x + 2)/(-1x + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 40 | math |
trans_func_intersection_train_0718 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2(-1e^(2x) + 2)^2 - 1(-1e^(2x) + 2) + 2 and g(x) = -1|x + 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0719 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 1((-1(-4x - 5) - 2)/(1(-4x - 5) + 3))^2 + 3((-1(-4x - 5) - 2)/(1(-4x - 5) + 3)) - 5 and g(x) = 2|(-5x - 5) + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0720 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = (2(3|(4x - 3)| - 3) + 2)/(3(3|(4x - 3)| - 3) + 1) and g(x) = (-1(5x - 2) + 3)/(-2(5x - 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0721 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 2|(-3sin(2πx + 3) + 1) - 3| - 1 = 4x - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
trans_func_intersection_train_0722 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 1|(2cos(2π((-1(2x - 1))/(1(2x - 1) - 2)) - 1) + 1) + 3| = -5((1(-1|x + 3| + 1))/(2(-1|x + 3| + 1) + 2)) + 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_0723 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (3(-1(-3|(2x^2 + 1x + 2) + 2| - 1) - 5) - 3)/(-3(-1(-3|(2x^2 + 1x + 2) + 2| - 1) - 5) - 1) and g(x) = -3(2((1x + 3)/(-3x - 1))^2 - 1((1x + 3)/(-3x - 1)) - 5) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer i... | 2 | math |
trans_func_intersection_train_0724 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -4(-1|((1(-3x^2 + 3x - 4) + 3)/(2(-3x^2 + 3x - 4) - 1)) + 3|) + 5 and g(x) = (1(-5(2|x + 2| - 2) + 1) + 3)/(1(-5(2|x + 2| - 2) + 1) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Y... | 2 | math |
trans_func_intersection_train_0725 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1(3e^(1(-3|((-1x + 2)/(-1x)) - 1| - 2)) - 2) and g(x) = (-1(2(-1x^2 + 3x + 3) - 4))/(2(2(-1x^2 + 3x + 3) - 4) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0726 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -5(-2sin(2π(-1|x + 1| - 3) - 3) - 1) - 3 and g(x) = 3sin(1π(-3x - 2) - 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 42 | math |
trans_func_intersection_train_0727 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3cos(2π(1(-4(-3|x + 3| + 2) - 4)^2 + 5(-4(-3|x + 3| + 2) - 4) - 2) - 1) and g(x) = -2sin(2π((-1(1|x - 2| - 2) + 2)/(-3(1|x - 2| - 2) - 2)) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{You... | 1423 | math |
trans_func_intersection_train_0728 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 1|(4(2sin(3π(-5x - 1) + 3) + 2)^2 - 3) + 2| - 2 = (2(2(-3sin(3πx - 2)) - 2) - 3)/(-1(2(-3sin(3πx - 2)) - 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 163 | math |
trans_func_intersection_train_0729 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -4((1(2(3|x| + 2)^2 - 5(3|x| + 2) + 1) + 3)/(1(2(3|x| + 2)^2 - 5(3|x| + 2) + 1))) + 4 and g(x) = -1e^(2(5(3|x| - 1)^2 - 3(3|x| - 1) - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your a... | 4 | math |
trans_func_intersection_train_0730 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-3(-2(3|(-1x + 5)| + 1)^2 + 3(3|(-1x + 5)| + 1) + 5) + 2)/(-1(-2(3|(-1x + 5)| + 1)^2 + 3(3|(-1x + 5)| + 1) + 5) + 2) and g(x) = -2cos(2π((3(1|x - 3| + 2) - 1)/(-1(1|x - 3| + 2) - 3)) - 2) - 2 in the range -10 ≤... | 2 | math |
trans_func_intersection_train_0731 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 1cos(1π(3|x - 3| + 3) - 1) + 3 and g(x) = (-3x - 1)/(-2x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
trans_func_intersection_train_0732 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (1(-3(1|(-5x^2 - 1x - 4) + 1|) + 1) - 2)/(-2(-3(1|(-5x^2 - 1x - 4) + 1|) + 1) - 2) = 5(2sin(3π((-3x - 1)/(2x)) + 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 47 | math |
trans_func_intersection_train_0733 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (3(3|x - 1| - 1))/(3(3|x - 1| - 1) + 3) and g(x) = -2|x - 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0734 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (2(1(1(1|x + 2| - 2) - 2)^2 + 2(1(1|x + 2| - 2) - 2) + 1))/(-1(1(1(1|x + 2| - 2) - 2)^2 + 2(1(1|x + 2| - 2) - 2) + 1) + 2) and g(x) = (-3(3e^(-2(2|x| - 2))) + 2)/(1(3e^(-2(2|x| - 2))) - 2) in the range -10 ≤ x ≤... | 2 | math |
trans_func_intersection_train_0735 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 4(-3((3x - 3)/(-2x - 2)) - 1)^2 - 5(-3((3x - 3)/(-2x - 2)) - 1) + 1 and g(x) = 4(-3x + 5)^2 - 2(-3x + 5) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0736 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 2sin(3π(-4(-1|(5x - 1) + 2| - 1)^2 + 4(-1|(5x - 1) + 2| - 1) + 3) + 1) - 2 and g(x) = -5(1|((-1x + 2)/(2x + 2)) - 3| - 1)^2 + 1(1|((-1x + 2)/(2x + 2)) - 3| - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex... | 29 | math |
trans_func_intersection_train_0737 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -2((-2(-2|x - 2| - 1) + 1)/(2(-2|x - 2| - 1) - 1)) - 2 and g(x) = (2(-1|x + 1| + 1) - 1)/(1(-1|x + 1| + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0738 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 3(3cos(3π(-2x^2 + 2) - 2) - 3) + 1 and g(x) = -1|(3sin(2πx + 1) + 1) - 1| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 970 | math |
trans_func_intersection_train_0739 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (1(3|(-3(-2x^2 - 5x - 2) - 5) + 3| + 1) + 3)/(-3(3|(-3(-2x^2 - 5x - 2) - 5) + 3| + 1) + 1) = 3sin(2π(4(-5x^2 - 1x - 2) - 4) - 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1433 | math |
trans_func_intersection_train_0740 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (3(-1cos(3πx - 1) - 1) + 1)/(-1(-1cos(3πx - 1) - 1) + 2) and g(x) = -2|x + 1| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0741 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1sin(1π(-3(1|x - 1|) + 3) + 1) - 2 and g(x) = 2(2|x + 3|)^2 - 2(2|x + 3|) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0742 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 4((-1(-1(1|x - 3| + 1)^2 + 2(1|x - 3| + 1) + 1))/(3(-1(1|x - 3| + 1)^2 + 2(1|x - 3| + 1) + 1) + 2)) - 5 and g(x) = -1|(-1(1x - 2)^2 + 2(1x - 2) - 4) - 3| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in L... | 2 | math |
trans_func_intersection_train_0743 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 4(-2|(-2e^(1x) + 2) - 1|) + 5 and g(x) = -3sin(1π(-5x^2 - 1x - 1) - 3) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 479 | math |
trans_func_intersection_train_0744 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -3(-3|(-2sin(1π(-4x + 2)) + 2) - 1| + 2)^2 - 3(-3|(-2sin(1π(-4x + 2)) + 2) - 1| + 2) + 1 and g(x) = -3(2((1x + 1)/(1x + 2)))^2 + 1(2((1x + 1)/(1x + 2))) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in La... | 198 | math |
trans_func_intersection_train_0745 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -3|(2sin(3πx + 2) + 1) - 2| - 3 = 1x^2 + 4x have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 12 | math |
trans_func_intersection_train_0746 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 1cos(2π(-1x + 3) + 3) + 2 and g(x) = -1x^2 - 2x + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0747 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (2(3(5x + 5)^2 + 3(5x + 5)) - 3)/(-3(3(5x + 5)^2 + 3(5x + 5)) + 2) and g(x) = -3e^(-1((2x - 2)/(3x + 1))) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0748 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 3|(1(3x^2 + 2) + 3)| - 3 and g(x) = 3|(5x - 2)| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0749 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -3sin(1π(-5(-1|((3x + 1)/(2x - 1)) - 2| + 1)^2 - 4(-1|((3x + 1)/(2x - 1)) - 2| + 1) + 2)) and g(x) = 1|(1cos(2π((1x + 3)/(1x + 2))) - 1) - 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: ... | 15 | math |
trans_func_intersection_train_0750 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -3((-1(1|(4x + 3) - 2| + 2) + 3)/(-2(1|(4x + 3) - 2| + 2) - 2))^2 - 3((-1(1|(4x + 3) - 2| + 2) + 3)/(-2(1|(4x + 3) - 2| + 2) - 2)) and g(x) = 3((1(-4x^2) + 3)/(1(-4x^2))) + 1 where -10 ≤ x ≤ 10.\n\nPresent... | 2 | math |
trans_func_intersection_train_0751 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (2(4(1(-2|x + 2| + 3)^2 + 4(-2|x + 2| + 3) - 3) - 4) - 2)/(-3(4(1(-2|x + 2| + 3)^2 + 4(-2|x + 2| + 3) - 3) - 4)) = 3sin(1π(-5((2x + 3)/(-2x - 3))^2 + 3((2x + 3)/(-2x - 3)) - 3) - 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the an... | 4 | math |
trans_func_intersection_train_0752 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 1(-1cos(2πx - 2))^2 - 5(-1cos(2πx - 2)) - 3 and g(x) = 4x^2 - 2x - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
trans_func_intersection_train_0753 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (2(3|(2cos(3πx + 2) + 1) - 3|) - 3)/(3(3|(2cos(3πx + 2) + 1) - 3|) + 3) and g(x) = -1cos(3π(-3x^2 + 2) - 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0754 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -1sin(1π((-1(1x^2 + 1x + 3) - 2)/(1(1x^2 + 1x + 3))) - 1) - 2 and g(x) = 5(4x + 5)^2 + 4(4x + 5) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0755 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 2|(-1x^2 + 2x - 2)| - 2 and g(x) = 3x^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_0756 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -3sin(3π(4x + 1)) - 1 and g(x) = 1x - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 71 | math |
trans_func_intersection_train_0757 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (2(-3cos(3πx)))/(3(-3cos(3πx)) + 3) = -3|x - 3| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
trans_func_intersection_train_0758 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 5(-3cos(1πx + 2) - 3)^2 + 2(-3cos(1πx + 2) - 3) and g(x) = (-1x - 1)/(-2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 19 | math |
trans_func_intersection_train_0759 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2(1|(2(2sin(3πx - 3) + 2)^2 - 1(2sin(3πx - 3) + 2) - 1)| + 1) + 2 and g(x) = -5(-1sin(2π(-3|x - 2| + 2) - 2) + 1) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
... | 120 | math |
trans_func_intersection_train_0760 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1((-2(-1sin(3π(-1x^2 + 4x - 3) - 2)) - 3)/(3(-1sin(3π(-1x^2 + 4x - 3) - 2)) - 1)) + 2 and g(x) = -1e^(1((-1(3|x - 3| - 3) - 1)/(1(3|x - 3| - 3) + 2))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: ... | 407 | math |
trans_func_intersection_train_0761 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2|(-2sin(3π(-3((1x - 2)/(1x)) - 3) - 2) + 2) - 3| + 2 and g(x) = (-1(4(1cos(2πx + 1) + 3)^2 - 2(1cos(2πx + 1) + 3) + 5))/(3(4(1cos(2πx + 1) + 3)^2 - 2(1cos(2πx + 1) + 3) + 5) + 3) where -10 ≤ x ≤ 10.\n\nP... | 146 | math |
trans_func_intersection_train_0762 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 3sin(2π(-2|x - 3| + 2) + 1) + 1 and g(x) = 1x^2 + 4x - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 8 | math |
trans_func_intersection_train_0763 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -1|(1((-1x)/(-1x + 2))^2 + 3((-1x)/(-1x + 2)) - 2) - 1| + 3 and g(x) = (2(1x^2 + 3x + 4) + 1)/(-2(1x^2 + 3x + 4) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0764 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 4((3(2cos(1πx - 1)) + 2)/(2(2cos(1πx - 1)) - 1))^2 + 3((3(2cos(1πx - 1)) + 2)/(2(2cos(1πx - 1)) - 1)) - 4 and g(x) = (-3(3x^2 - 3x - 3) - 1)/(-1(3x^2 - 3x - 3) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answe... | 24 | math |
trans_func_intersection_train_0765 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -2(-3|x - 1| - 1)^2 - 4(-3|x - 1| - 1) + 3 and g(x) = 2x - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0766 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 5(4x^2 - 3x - 1) + 5 and g(x) = (-3x)/(-1x + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
trans_func_intersection_train_0767 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (1(-1(-2|x + 2| - 3) - 3) + 3)/(3(-1(-2|x + 2| - 3) - 3) + 1) and g(x) = 1cos(1π((3x - 2)/(-2x + 1)) + 1) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0768 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-1(4(3|x - 3| - 2) - 2) + 1)/(-1(4(3|x - 3| - 2) - 2) - 3) and g(x) = 3cos(3π(-2x)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 118 | math |
trans_func_intersection_train_0769 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -1e^(1(1|x - 2| - 3)) and g(x) = -5x + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0770 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 4((-1(1e^(-1x) + 1) + 2)/(2(1e^(-1x) + 1) - 3)) + 1 and g(x) = -1sin(2π((-1x - 2)/(-1x - 1)) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 48 | math |
trans_func_intersection_train_0771 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 3cos(1π(-1(2|((1x + 1)/(3x - 2))| - 3)^2 - 3) + 2) + 2 = -4((-2(-1sin(3πx - 3) + 1))/(-3(-1sin(3πx - 3) + 1) + 2))^2 - 1((-2(-1sin(3πx - 3) + 1))/(-3(-1sin(3πx - 3) + 1) + 2)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer i... | 102 | math |
trans_func_intersection_train_0772 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -4((3(-5x - 2) + 1)/(1(-5x - 2) + 1))^2 + 2((3(-5x - 2) + 1)/(1(-5x - 2) + 1)) + 1 and g(x) = -2|(-3x + 5) + 1| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0773 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation (-2(-3x^2 - 1) - 1)/(-3(-3x^2 - 1) + 1) = 2|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_0774 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation 3(-2e^(-1x) + 3)^2 + 5(-2e^(-1x) + 3) - 1 = -2x^2 - 1x + 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_0775 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1|(1((-1(3x^2 + 1x) - 1)/(2(3x^2 + 1x) + 1)) - 2) - 1| - 1 and g(x) = -3((-2(2|x - 2| - 1) + 1)/(2(2|x - 2| - 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_0776 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 3sin(2π(1(-4(1|x + 2|) + 3)^2 - 4(-4(1|x + 2|) + 3) + 3) + 3) and g(x) = (-1(-2|(-2sin(1πx + 1) - 2) + 1|) + 2)/(2(-2|(-2sin(1πx + 1) - 2) + 1|) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex form... | 905 | math |
trans_func_intersection_train_0777 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 3sin(1π(2x^2 + 1x - 5) + 1) and g(x) = (1x - 3)/(2x + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 397 | math |
trans_func_intersection_train_0778 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -2sin(2π((-3(1(2x^2 - 2) - 1) - 2)/(1(1(2x^2 - 2) - 1) - 3)) + 3) + 2 = 1((-1(-2cos(1πx + 1)) - 2)/(1(-2cos(1πx + 1)))) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 62 | math |
trans_func_intersection_train_0779 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -3((2(-2|x - 2| + 3) + 3)/(1(-2|x - 2| + 3) + 3)) - 3 and g(x) = 1(-4x^2 - 2x) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0780 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -2(-1x + 1)^2 - 4 and g(x) = -2|x + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0781 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (-1(3|(5x^2 + 2x - 1) - 3| - 2))/(-1(3|(5x^2 + 2x - 1) - 3| - 2) + 2) and g(x) = -2|((2x + 3)/(3x - 1)) - 3| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
trans_func_intersection_train_0782 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 1cos(1π(-2x^2 - 5x) + 2) - 1 and g(x) = (3x + 1)/(-2x + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 371 | math |
trans_func_intersection_train_0783 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-2(-4(-3(-1|x + 1| - 1) - 2)^2 + 2(-3(-1|x + 1| - 1) - 2) + 5) + 3)/(-2(-4(-3(-1|x + 1| - 1) - 2)^2 + 2(-3(-1|x + 1| - 1) - 2) + 5) - 1) and g(x) = 2|(-2((1x + 2)/(-1x))^2 + 1) - 1| - 1 in the range -10 ≤ x ≤ 1... | 2 | math |
trans_func_intersection_train_0784 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1(-3|(2x^2 - 3x - 3) - 2| + 2) - 1 = 1(-4x - 3)^2 + 1(-4x - 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_0785 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -4(-5((2(1|x| - 3) - 1)/(-2(1|x| - 3) + 3))^2 - 4) and g(x) = (-3(-3|(-5x + 5) + 2| + 3) - 3)/(-1(-3|(-5x + 5) + 2| + 3) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 2 | math |
trans_func_intersection_train_0786 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 5(-3(-2|(-3sin(3πx - 1)) + 3| + 2) - 1)^2 - 5 and g(x) = (1(-1|(-1e^(1x) + 3) - 3| - 2) + 1)/(2(-1|(-1e^(1x) + 3) - 3| - 2) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 120 | math |
trans_func_intersection_train_0787 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -3sin(3π((2(3x + 5) - 2)/(3(3x + 5) - 2))) - 1 and g(x) = (-1(2sin(2πx + 2) - 2) - 2)/(-2(2sin(2πx + 2) - 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 48 | math |
trans_func_intersection_train_0788 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = (2(-1(-2x^2 + 3x - 2) - 5) - 3)/(-1(-1(-2x^2 + 3x - 2) - 5) - 2) and g(x) = -2cos(2π(1x + 4) + 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
trans_func_intersection_train_0789 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -1((2(3x^2 - 4x) + 2)/(-3(3x^2 - 4x) - 1)) + 1 = 2e^(-2(4x)) - 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_0790 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2(-1cos(3π(3|x + 3| + 3) + 3) + 1)^2 + 4(-1cos(3π(3|x + 3| + 3) + 3) + 1) + 5 and g(x) = 1|((-3x + 3)/(-2x + 3)) + 2| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
... | 4 | math |
trans_func_intersection_train_0791 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -2(2(-2sin(3πx - 3) - 2) + 2)^2 + 5(2(-2sin(3πx - 3) - 2) + 2) + 3 = 3sin(1π(-1x + 3)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 60 | math |
trans_func_intersection_train_0792 | trans_func_intersection | null | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 1|(4x - 3) + 2| + 1 and g(x) = 4x^2 - 5x where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0793 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -1|(1(2sin(1π((-1x + 2)/(-2x + 3)) - 1) - 2)^2 - 5(2sin(1π((-1x + 2)/(-2x + 3)) - 1) - 2) - 5) + 2| - 2 and g(x) = -2cos(3π(2|((-1x - 3)/(2x - 1)) + 1| + 3) + 2) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the ans... | 80 | math |
trans_func_intersection_train_0794 | trans_func_intersection | null | [
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -5(-1((2(-1|x - 2| + 1) - 2)/(-3(-1|x - 2| + 1) - 3)) + 3)^2 - 1(-1((2(-1|x - 2| + 1) - 2)/(-3(-1|x - 2| + 1) - 3)) + 3) + 4 and g(x) = -3|(1(-3e^(1x) + 1) + 1) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in ... | 5 | math |
trans_func_intersection_train_0795 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1((2(-4(-3|x - 2|) - 5) - 2)/(-3(-4(-3|x - 2|) - 5)))^2 - 1((2(-4(-3|x - 2|) - 5) - 2)/(-3(-4(-3|x - 2|) - 5))) + 3 and g(x) = -1sin(2π(2(-3|x + 1| + 2)^2 + 3(-3|x + 1| + 2) + 1)) + 3 in the range -10 ≤ x ≤ 10?\... | 116 | math |
trans_func_intersection_train_0796 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-1(-3sin(3π(4x + 2)) + 2) - 3)/(2(-3sin(3π(4x + 2)) + 2) - 3) and g(x) = -3((-3x)/(2x - 1))^2 + 5((-3x)/(2x - 1)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
... | 83 | math |
trans_func_intersection_train_0797 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many solutions does the equation -2(-4x^2 + 5x + 2) - 2 = -4x^2 + 5x + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
trans_func_intersection_train_0798 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2((-2(3e^(1x)))/(-2(3e^(1x)) - 3))^2 + 4((-2(3e^(1x)))/(-2(3e^(1x)) - 3)) + 2 and g(x) = 2sin(3π(5x - 4) - 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 190 | math |
trans_func_intersection_train_0799 | trans_func_intersection | null | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 5(-5((-2x - 2)/(-1x - 2)) + 3)^2 + 4(-5((-2x - 2)/(-1x - 2)) + 3) - 2 and g(x) = -3(-2e^(2x) - 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
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