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[ { "content": "Find the number of times the graphs of f(x) = -2cos(2π(-4((-2(-1|x + 1|))/(-1(-1|x + 1|) + 2)) - 2)) + 1 and g(x) = (3(-2e^(-1((-1x + 3)/(2x - 1))) - 1) - 3)/(1(-2e^(-1((-1x + 3)/(2x - 1))) - 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",...
26
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -3(3(-3((-1x + 1)/(3x + 1))^2 + 5((-1x + 1)/(3x + 1)) + 5)^2 - 1(-3((-1x + 1)/(3x + 1))^2 + 5((-1x + 1)/(3x + 1)) + 5) - 1)^2 + 2(3(-3((-1x + 1)/(3x + 1))^2 + 5((-1x + 1)/(3x + 1)) + 5)^2 - 1(-3((-1x + 1)/(3x + 1))^2 + 5((-1x...
4
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = (-3(5(2sin(3π(-2x + 4)) + 1)) + 1)/(3(5(2sin(3π(-2x + 4)) + 1)) + 3) and g(x) = 1(-1((2x - 1)/(-1x - 2)) - 5)^2 + 3(-1((2x - 1)/(-1x - 2)) - 5) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answe...
49
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -4((3((1(-1|x + 1| + 2) + 2)/(-1(-1|x + 1| + 2) + 3)) + 3)/(3((1(-1|x + 1| + 2) + 2)/(-1(-1|x + 1| + 2) + 3)) + 1)) - 3 and g(x) = -2|(5(-2|x - 2| - 1)^2 + 5(-2|x - 2| - 1) - 2) - 3| + 2 where -10 ≤ x ≤ 10.\n\nPresent the ans...
4
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 2|(2sin(2π(-3|(5x^2 - 4x + 2) - 1| + 3) - 1) + 3) + 2| - 2 and g(x) = 2(2|((2x - 2)/(-3x - 1))| + 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
1101
MATH
[ { "content": "How many solutions does the equation (-2(-2(-3cos(1π(-2x + 5))) + 3) - 2)/(2(-2(-3cos(1π(-2x + 5))) + 3) + 3) = 2|((-2(1e^(-1x) - 1) + 3)/(-3(1e^(-1x) - 1) + 1)) - 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
39
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -4(-2cos(3π(-5(-1|x + 1| + 3)^2 + 3(-1|x + 1| + 3) + 4) + 3))^2 - 2(-2cos(3π(-5(-1|x + 1| + 3)^2 + 3(-1|x + 1| + 3) + 4) + 3)) + 2 and g(x) = (-3(-3|(-5x^2 + 5x + 3)| + 3))/(-3(-3|(-5x^2 + 5x + 3)| + 3) - 1) in the range -10 ≤ x ≤ ...
863
MATH
[ { "content": "How many solutions does the equation 3(1(5(5x + 5) + 5) + 3)^2 + 4(1(5(5x + 5) + 5) + 3) = -1sin(1π(-2(5x - 2)^2 - 5(5x - 2) - 1) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
4
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = (-3(-3(-1(3cos(2πx + 2) - 3)^2 - 1(3cos(2πx + 2) - 3) + 1) - 1) + 3)/(3(-3(-1(3cos(2πx + 2) - 3)^2 - 1(3cos(2πx + 2) - 3) + 1) - 1) - 1) and g(x) = (3(5(-2x^2 - 4x + 1) + 1) + 3)/(-1(5(-2x^2 - 4x + 1) + 1) + 3) in the range -10 ≤ x...
27
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -3|(-3|((-3(5x^2 + 5x + 1) + 1)/(1(5x^2 + 5x + 1) + 2)) + 2| - 1) + 1| + 3 and g(x) = -2sin(3π((1(5x + 5) - 2)/(3(5x + 5) + 3))) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role...
13
MATH
[ { "content": "Find the number of times the graphs of f(x) = 3(5((2(-3x - 2) - 1)/(1(-3x - 2) + 1))^2 + 1((2(-3x - 2) - 1)/(1(-3x - 2) + 1)) - 1)^2 - 1(5((2(-3x - 2) - 1)/(1(-3x - 2) + 1))^2 + 1((2(-3x - 2) - 1)/(1(-3x - 2) + 1)) - 1) - 5 and g(x) = -1(3e^(1(-2|x - 1| - 3)) - 1) - 3 intersect in the interval [-1...
4
MATH
[ { "content": "Find the number of times the graphs of f(x) = -2(-2(3|((2x + 1)/(1x + 1)) - 3| + 3) + 1)^2 + 5(-2(3|((2x + 1)/(1x + 1)) - 3| + 3) + 1) + 5 and g(x) = -4(5(-1cos(3πx)) + 3)^2 - 1(5(-1cos(3πx)) + 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"...
26
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -2((2((1(-1x^2 + 1x + 2) + 3)/(3(-1x^2 + 1x + 2) - 2)) + 3)/(-2((1(-1x^2 + 1x + 2) + 3)/(3(-1x^2 + 1x + 2) - 2))))^2 - 2((2((1(-1x^2 + 1x + 2) + 3)/(3(-1x^2 + 1x + 2) - 2)) + 3)/(-2((1(-1x^2 + 1x + 2) + 3)/(3(-1x^2 + 1x + 2) ...
6
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -1(-2|(3|(-1|x - 1| - 1) + 1| - 2) - 3| + 1)^2 + 4(-2|(3|(-1|x - 1| - 1) + 1| - 2) - 3| + 1) + 5 and g(x) = -1|(1|((-2x + 3)/(2x - 1)) - 1| + 3) - 1| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your...
6
MATH
[ { "content": "How many solutions does the equation 1e^(2(-2(4((2x)/(2x - 1)) + 1) - 4)) - 1 = (-1(-3e^(-2(-3x + 1)) - 3) + 2)/(1(-3e^(-2(-3x + 1)) - 3) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
34
MATH
[ { "content": "Find the number of times the graphs of f(x) = -2|(3(3(2|x + 1| - 3) - 4) - 3) - 2| - 1 and g(x) = 2sin(3π(3(3x - 3)^2 + 3(3x - 3) - 3) - 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
22
MATH
[ { "content": "Find the number of times the graphs of f(x) = 3|(-1|(1|(1|x + 1| - 3) - 2|) + 2| + 1) + 2| + 1 and g(x) = -5(-2((-3x)/(2x + 1)) - 3)^2 - 5(-2((-3x)/(2x + 1)) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
4
MATH
[ { "content": "Find the number of times the graphs of f(x) = -4(1(-2cos(1π(-2x^2 - 5x + 5) + 1) - 1)^2 - 5(-2cos(1π(-2x^2 - 5x + 5) + 1) - 1) - 1)^2 + 4(1(-2cos(1π(-2x^2 - 5x + 5) + 1) - 1)^2 - 5(-2cos(1π(-2x^2 - 5x + 5) + 1) - 1) - 1) - 2 and g(x) = -3|((-1(3|x - 3|) - 3)/(-1(3|x - 3|) + 3)) + 3| + 1 intersect ...
500
MATH
[ { "content": "Find the number of times the graphs of f(x) = 3|(-3cos(1π(-3|(1x^2 + 2x - 5) - 2| - 1) - 1) + 2) - 1| + 1 and g(x) = 5(2|(2sin(2πx) - 2) - 2| - 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
176
MATH
[ { "content": "How many solutions does the equation -1|(-3((2(3|x + 3| + 2) - 2)/(3(3|x + 3| + 2) + 2))^2 + 3((2(3|x + 3| + 2) - 2)/(3(3|x + 3| + 2) + 2))) + 3| - 1 = -5(2sin(2π(1x^2 + 1x + 3) - 1) + 2) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "r...
376
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -5(-1e^(2(1(-3x^2 - 5x + 5) - 1)) + 1)^2 - 2(-1e^(2(1(-3x^2 - 5x + 5) - 1)) + 1) - 1 and g(x) = -1(1(3x)^2 + 4) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
4
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -4((-3(-2((2x)/(-2x - 1))^2 - 1((2x)/(-2x - 1))) - 2)/(-1(-2((2x)/(-2x - 1))^2 - 1((2x)/(-2x - 1))) + 2))^2 + 1((-3(-2((2x)/(-2x - 1))^2 - 1((2x)/(-2x - 1))) - 2)/(-1(-2((2x)/(-2x - 1))^2 - 1((2x)/(-2x - 1))) + 2)) + 5 and g(...
4
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -1|(1sin(1π(1((-3x + 3)/(-1x + 2)) + 5) + 3) - 1) - 2| + 2 and g(x) = 3(-1(1x^2 - 2x - 5) - 3) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
6
MATH
[ { "content": "Find the number of times the graphs of f(x) = 5(3cos(2π(-2(2x - 2)^2 - 2(2x - 2) + 3)) + 1) - 2 and g(x) = (-1((2(-2|x - 2| + 2) - 3)/(3(-2|x - 2| + 2) - 2)) + 2)/(-3((2(-2|x - 2| + 2) - 3)/(3(-2|x - 2| + 2) - 2)) + 4) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\b...
1646
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 3cos(3π(5((2(-3|x - 1| + 1) - 1)/(-1(-3|x - 1| + 1) - 2))) + 3) - 2 and g(x) = 1(5(-3|x| + 2) - 4)^2 - 2(5(-3|x| + 2) - 4) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "use...
48
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -4(1(-1|(-1x - 4) + 1| + 1)^2 + 4(-1|(-1x - 4) + 1| + 1) + 2)^2 + 3(1(-1|(-1x - 4) + 1| + 1)^2 + 4(-1|(-1x - 4) + 1| + 1) + 2) and g(x) = -2((-3(2sin(1πx) - 2) + 2)/(-1(2sin(1πx) - 2) + 2))^2 + 3((-3(2sin(1πx) - 2) + 2)/(-1(2...
8
MATH
[ { "content": "Find the number of times the graphs of f(x) = (2(-1cos(2π((2((-1x + 3)/(1x - 2)) - 3)/(2((-1x + 3)/(1x - 2)) - 1))) + 2))/(3(-1cos(2π((2((-1x + 3)/(1x - 2)) - 3)/(2((-1x + 3)/(1x - 2)) - 1))) + 2)) and g(x) = 1(-3sin(3π(-2|x + 3| + 3) - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the a...
120
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = 5(3|(-3|(2cos(1πx) - 3)| + 2) + 3| + 3)^2 + 3(3|(-3|(2cos(1πx) - 3)| + 2) + 3| + 3) and g(x) = 2(-3|(-4x^2 + 2x + 3) - 1| + 1)^2 + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role...
4
MATH
[ { "content": "Find the number of times the graphs of f(x) = -3(2cos(3π(-1|(3x + 2) + 3| - 2) + 1)) + 3 and g(x) = (-3(1sin(1π(-2x^2 - 4x + 4) - 1) + 1) - 2)/(2(1sin(1π(-2x^2 - 4x + 4) - 1) + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "u...
253
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -1(-5(-2(-1e^(-1x) + 3)^2 + 3(-1e^(-1x) + 3) + 2)^2) - 1 and g(x) = 3|((-1(1cos(2πx - 2) - 2) - 3)/(-2(1cos(2πx - 2) - 2))) - 1| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role"...
4
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 3(-3|(-3sin(3π(-1x^2 + 5x - 4)) + 3)| + 3) + 2 and g(x) = 1(-5((2x - 2)/(2x))^2 - 5((2x - 2)/(2x)) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
624
MATH
[ { "content": "Find the number of times the graphs of f(x) = -1((1(2(-2|x| - 2)^2 - 2(-2|x| - 2) - 3) - 2)/(-1(2(-2|x| - 2)^2 - 2(-2|x| - 2) - 3) - 1))^2 - 1((1(2(-2|x| - 2)^2 - 2(-2|x| - 2) - 3) - 2)/(-1(2(-2|x| - 2)^2 - 2(-2|x| - 2) - 3) - 1)) + 4 and g(x) = -2(-1((-2x - 2)/(1x - 3))^2 + 5((-2x - 2)/(1x - 3)) ...
4
MATH
[ { "content": "How many solutions does the equation -2sin(3π(-1|(-1|(2|x - 3| + 1) + 2| + 3) - 2| - 3)) - 3 = (-3(-1(-3|x| + 2) - 5) - 2)/(3(-1(-3|x| + 2) - 5)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
92
MATH
[ { "content": "How many solutions does the equation -1(-4(-5((-2x - 1)/(-1x)) + 1)^2 - 5(-5((-2x - 1)/(-1x)) + 1) + 5)^2 - 5(-4(-5((-2x - 1)/(-1x)) + 1)^2 - 5(-5((-2x - 1)/(-1x)) + 1) + 5) + 4 = 2cos(3π(-2|(-2x + 2) - 1| - 3) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \...
4
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -3((-2(4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 3) - 3)/(1(4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 3)))^2 - 5((-2(4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 3) - 3)/(1(4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 3))) + 1 and g(x) = (2(1(4x^2 + 3x ...
7
MATH
[ { "content": "Find the number of times the graphs of f(x) = -1((-3(-1|(1x^2 + 3x - 2)| + 1) - 2)/(-2(-1|(1x^2 + 3x - 2)| + 1) - 3))^2 - 3((-3(-1|(1x^2 + 3x - 2)| + 1) - 2)/(-2(-1|(1x^2 + 3x - 2)| + 1) - 3)) and g(x) = -2cos(1π(1|(-4x^2 + 5x - 4) - 2| - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the...
20
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -4(2sin(2π(2|((-2x - 1)/(2x - 1)) - 3| + 3) - 1) - 2)^2 - 5(2sin(2π(2|((-2x - 1)/(2x - 1)) - 3| + 3) - 1) - 2) - 4 and g(x) = -5(2|(-1cos(1πx + 3) - 3) + 1| + 2) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format:...
62
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = (1((1(1cos(1π(-5x^2 + 4x) + 3) - 1) + 2)/(-2(1cos(1π(-5x^2 + 4x) + 3) - 1) - 1)) - 3)/(3((1(1cos(1π(-5x^2 + 4x) + 3) - 1) + 2)/(-2(1cos(1π(-5x^2 + 4x) + 3) - 1) - 1)) - 2) and g(x) = 1|(-1(1x - 3) + 2) - 1| + 2 in the range -10 ≤ x...
210
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -3|(1sin(1π(-1(-2|x - 2| + 2)^2 + 3(-2|x - 2| + 2) + 1) + 3) + 1) + 1| + 1 and g(x) = -4(3(-5x^2 + 2x + 5) + 1)^2 + 1(3(-5x^2 + 2x + 5) + 1) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}...
4
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = 2|(5((-3(-2x - 4) - 2)/(3(-2x - 4) + 3)) + 5) + 3| + 3 and g(x) = 4(3|((2x + 2)/(1x + 2)) - 1| - 2) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
4
MATH
[ { "content": "How many solutions does the equation -1(-5(2|(-3x - 2) - 3| - 2)^2 + 4(2|(-3x - 2) - 3| - 2) - 5)^2 - 2(-5(2|(-3x - 2) - 3| - 2)^2 + 4(2|(-3x - 2) - 3| - 2) - 5) - 5 = -1|(3e^(2(-1|x + 3| + 3)))| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"...
4
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 1(2(1|((-2x - 3)/(-2x + 3)) - 2| - 1) - 4)^2 - 5 and g(x) = 1cos(1π((-3(2|x - 2| - 3))/(-1(2|x - 2| - 3) + 2)) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ...
8
MATH
[ { "content": "Find the number of times the graphs of f(x) = (-1(5(-3(1|x + 1|) + 2)^2 + 1(-3(1|x + 1|) + 2) - 5) - 3)/(2(5(-3(1|x + 1|) + 2)^2 + 1(-3(1|x + 1|) + 2) - 5) - 1) and g(x) = -1|(1cos(3π(2x^2 - 4x + 1) + 3) + 3) + 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\bo...
4
MATH
[ { "content": "How many solutions does the equation (2(2e^(1((-3(-1|x| - 3) - 1)/(-2(-1|x| - 3) + 1))) + 2) - 1)/(-2(2e^(1((-3(-1|x| - 3) - 1)/(-2(-1|x| - 3) + 1))) + 2) - 2) = 3|(-3sin(3π(-3|x| + 1) - 3)) - 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"...
360
MATH
[ { "content": "How many solutions does the equation (-3(1|(4(2sin(2πx - 2) - 2)^2 + 4(2sin(2πx - 2) - 2) + 4) + 2| + 1) + 1)/(1(1|(4(2sin(2πx - 2) - 2)^2 + 4(2sin(2πx - 2) - 2) + 4) + 2| + 1) + 1) = -2|((3(-3cos(3πx + 3) + 2) + 2)/(-1(-3cos(3πx + 3) + 2) + 2)) + 1| have for x in the interval [-10, 10]?\n\nPresen...
60
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = 1sin(2π(4(-2|((2x - 3)/(2x - 3)) - 3|) - 2)) - 2 and g(x) = -3|(2|(-2sin(2πx + 3) - 3)| - 2) - 1| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
80
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 5(-2cos(2π((2(2|x - 2| - 2) - 1)/(-3(2|x - 2| - 2) - 3)) - 2) + 3)^2 - 3(-2cos(2π((2(2|x - 2| - 2) - 1)/(-3(2|x - 2| - 2) - 3)) - 2) + 3) - 5 and g(x) = -2sin(2π(-3(2x^2 - 2x + 3)^2 - 3(2x^2 - 2x + 3) + 5) + 1) where -10 ≤ x ...
395
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = (-3(-3|(-1|((-2x + 1)/(1x + 1))| + 1) + 3| + 1) - 3)/(-1(-3|(-1|((-2x + 1)/(1x + 1))| + 1) + 3| + 1)) and g(x) = -1sin(3π(3(1x^2 - 5x - 5)^2 + 3(1x^2 - 5x - 5) + 3) - 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex...
138
MATH
[ { "content": "How many solutions does the equation (-2(4(5((-3x)/(3x + 3)) - 1) - 2) + 3)/(-3(4(5((-3x)/(3x + 3)) - 1) - 2)) = (-2(-5(2cos(3πx) - 2) + 2))/(-2(-5(2cos(3πx) - 2) + 2) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
57
MATH
[ { "content": "Find the number of times the graphs of f(x) = (1(2|(-2cos(2π(-3x + 3) + 3)) + 1|) + 3)/(-1(2|(-2cos(2π(-3x + 3) + 3)) + 1|) + 2) and g(x) = -1(-2(3cos(2πx)) - 1)^2 + 2(-2(3cos(2πx)) - 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "rol...
16
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 1sin(1π(2|(-2(5x^2 + 1x + 3)^2 - 3) - 2| + 2) - 2) and g(x) = -3(2e^(-1((3x - 3)/(-1x))) - 1) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
5
MATH
[ { "content": "How many solutions does the equation -5(3cos(1π((1(2x + 2) + 1)/(-2(2x + 2) - 3)) - 2) + 2) - 2 = -3cos(1π((-2(-3|x - 3|) + 2)/(-3(-3|x - 3|) + 2)) - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
10
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -1(3((3(2|x - 3| - 3) - 3)/(-3(2|x - 3| - 3) + 1)))^2 - 5 and g(x) = -2|(2|(3|x + 1| + 2) + 2| + 2) - 1| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
4
MATH
[ { "content": "How many solutions does the equation -5(2|(-3|(-2sin(2πx + 2) + 1) + 1|)| - 3)^2 = -3e^(1(1|((1x - 2)/(-1x - 1)) - 2| + 1)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
42
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = 2cos(3π(-5(-5(-2|x + 1|) + 1) - 3) + 1) + 2 and g(x) = 1(1(-3sin(2πx) - 3) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
727
MATH
[ { "content": "How many solutions does the equation -2e^(1((2(2(5x + 4) + 4) - 1)/(3(2(5x + 4) + 4) + 2))) + 2 = -1|(1(-2sin(1πx - 1)) - 3)| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
19
MATH
[ { "content": "Find the number of times the graphs of f(x) = 1(-5(1cos(2π(5x) - 2) - 1)^2 - 5(1cos(2π(5x) - 2) - 1) - 3)^2 + 5(-5(1cos(2π(5x) - 2) - 1)^2 - 5(1cos(2π(5x) - 2) - 1) - 3) + 2 and g(x) = 3cos(3π((1(-2|x - 3|) - 2)/(2(-2|x - 3|) + 1)) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answe...
219
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -3(3cos(1π((-2(3x) + 2)/(-3(3x) - 3)) + 3) + 1)^2 - 5(3cos(1π((-2(3x) + 2)/(-3(3x) - 3)) + 3) + 1) + 2 and g(x) = 1|(2(1cos(2πx + 2) + 3) - 4) + 1| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your a...
18
MATH
[ { "content": "How many solutions does the equation 4(-3sin(1π(4(-5x + 4) + 4) + 1) + 3) + 3 = -5(-4(-3e^(-1x) + 2) - 4) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
13
MATH
[ { "content": "Find the number of times the graphs of f(x) = 2|(-4(4((1x - 2)/(-2x - 1))^2 - 4)^2 - 5(4((1x - 2)/(-2x - 1))^2 - 4) + 4)| + 2 and g(x) = 4(-4(-1|x - 1| - 3) + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
4
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 5(2(-1(-2cos(3πx) - 2)^2 + 5)^2 + 1(-1(-2cos(3πx) - 2)^2 + 5) - 2) and g(x) = -3((-1((-3x - 3)/(3x - 2)) - 1)/(2((-3x - 3)/(3x - 2)) + 1)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", ...
119
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 1(3(-3|(-1x^2 - 1x - 2) + 2| - 2) - 4) - 4 and g(x) = -5(2cos(3π(-2x - 2) - 2) + 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
6
MATH
[ { "content": "Find the number of times the graphs of f(x) = 2|(1(-1((-2x + 3)/(-2x)) - 3) + 5)| - 2 and g(x) = 4(2sin(1π(3|x + 2| - 1)) + 2)^2 + 5(2sin(1π(3|x + 2| - 1)) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
52
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -5(-2|((-1(1cos(3πx - 1) + 2) + 1)/(-2(1cos(3πx - 1) + 2) + 3)) + 2| + 3) + 2 and g(x) = -4(1(1sin(1πx + 2) - 3)^2 + 2(1sin(1πx + 2) - 3) - 1)^2 + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answ...
50
MATH
[ { "content": "How many solutions does the equation 5(-4((1(-1x^2 - 5x + 5))/(2(-1x^2 - 5x + 5) - 1))^2 + 5) = -3(5(1cos(2πx + 1))^2 - 2(1cos(2πx + 1)) + 4) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
4
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = (2(3(3(-1x - 5) - 5)^2 + 1(3(-1x - 5) - 5) - 3) + 2)/(1(3(3(-1x - 5) - 5)^2 + 1(3(-1x - 5) - 5) - 3) + 3) and g(x) = 1|(-2|(2cos(2πx - 3) - 1)| - 3) + 2| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed...
40
MATH
[ { "content": "How many solutions does the equation -4(3|((-2(-1|x + 3| + 3))/(-2(-1|x + 3| + 3) + 3)) - 1| - 3)^2 - 3(3|((-2(-1|x + 3| + 3))/(-2(-1|x + 3| + 3) + 3)) - 1| - 3) - 3 = -3(2|(1sin(3πx) - 2) - 1| - 3)^2 + 4(2|(1sin(3πx) - 2) - 1| - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in La...
50
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = (-2(-5(2(-1x^2 + 1x) - 5)^2 + 1) + 1)/(2(-5(2(-1x^2 + 1x) - 5)^2 + 1) - 1) and g(x) = -4(2|(1|x - 2| - 3) - 1| - 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }...
4
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -4((-3(5(1|x - 2| - 1) + 1) - 2)/(-2(5(1|x - 2| - 1) + 1) - 1)) - 5 and g(x) = -3((2(2sin(3πx + 3) + 1) - 2)/(-3(2sin(3πx + 3) + 1) + 4)) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", ...
60
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -5(3sin(1π(3(3|x + 2| - 3) + 2) - 2) - 2)^2 - 3(3sin(1π(3(3|x + 2| - 3) + 2) - 2) - 2) - 1 and g(x) = -5((1(1x + 5) + 2)/(2(1x + 5) + 2))^2 + 1((1(1x + 5) + 2)/(2(1x + 5) + 2)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer i...
348
MATH
[ { "content": "Find the number of times the graphs of f(x) = -1(-4(-2cos(2π(-3|x - 3| - 1)) - 1)^2 + 5(-2cos(2π(-3|x - 3| - 1)) - 1) + 1)^2 + 1(-4(-2cos(2π(-3|x - 3| - 1)) - 1)^2 + 5(-2cos(2π(-3|x - 3| - 1)) - 1) + 1) - 1 and g(x) = -1e^(-2(4(1x^2 - 5x + 1) + 1)) - 1 intersect in the interval [-10, 10].\n\nPrese...
128
MATH
[ { "content": "Find the number of times the graphs of f(x) = -2sin(2π((3(4(-3|x - 3| + 2)^2 + 5(-3|x - 3| + 2) - 3))/(2(4(-3|x - 3| + 2)^2 + 5(-3|x - 3| + 2) - 3) + 1)) - 3) + 1 and g(x) = (-3(1|(-2|x - 3| + 1) + 1| - 1) - 2)/(-3(1|(-2|x - 3| + 1) + 1| - 1) - 3) intersect in the interval [-10, 10].\n\nPresent th...
36
MATH
[ { "content": "Find the number of times the graphs of f(x) = 1(-5(-3(4x^2 - 1x - 4) - 2)^2 + 5(-3(4x^2 - 1x - 4) - 2) + 2) + 2 and g(x) = -3((1(2x^2 + 2x - 5))/(-1(2x^2 + 2x - 5) - 2))^2 + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ...
6
MATH
[ { "content": "Find the number of times the graphs of f(x) = -5(-1(3(-3sin(1πx - 1) - 1)^2 - 1(-3sin(1πx - 1) - 1) - 5)^2 - 2(3(-3sin(1πx - 1) - 1)^2 - 1(-3sin(1πx - 1) - 1) - 5) + 3)^2 - 5(-1(3(-3sin(1πx - 1) - 1)^2 - 1(-3sin(1πx - 1) - 1) - 5)^2 - 2(3(-3sin(1πx - 1) - 1)^2 - 1(-3sin(1πx - 1) - 1) - 5) + 3) + 3...
139
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = (-1(-1((-1(3e^(1x) + 1) - 1)/(1(3e^(1x) + 1) + 2)) + 1) + 2)/(1(-1((-1(3e^(1x) + 1) - 1)/(1(3e^(1x) + 1) + 2)) + 1) + 3) and g(x) = 1((-3(1sin(3πx + 1) - 1))/(-2(1sin(3πx + 1) - 1) - 2))^2 + 4((-3(1sin(3πx + 1) - 1))/(-2(1sin...
60
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -3|((-1(-2cos(3π(-2|x + 2|) - 1) + 1) + 1)/(1(-2cos(3π(-2|x + 2|) - 1) + 1) - 2))| + 1 and g(x) = (-2((-2((1x + 3)/(2x)) - 1)/(-3((1x + 3)/(2x)) - 1)) + 1)/(2((-2((1x + 3)/(2x)) - 1)/(-3((1x + 3)/(2x)) - 1)) + 2) where -10 ≤ ...
236
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = (-3(-2cos(2π(1(-3x^2 + 2x + 1))) - 2) + 2)/(-1(-2cos(2π(1(-3x^2 + 2x + 1))) - 2) - 3) and g(x) = -3|(-4(-3|x - 2| - 2)^2 - 1(-3|x - 2| - 2) - 3) + 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your an...
17
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = -3(2cos(2π(3(-1|x| - 1) + 4) - 1) - 2) - 3 and g(x) = 1|(2|((-2x - 3)/(3x + 1)) + 3| - 1) - 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
118
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = 3(5((3((2x + 2)/(-2x - 1)))/(-3((2x + 2)/(-2x - 1)) - 1))^2 - 3((3((2x + 2)/(-2x - 1)))/(-3((2x + 2)/(-2x - 1)) - 1))) + 4 and g(x) = 4(-3|(1|x + 3| - 2) - 3|)^2 + 2(-3|(1|x + 3| - 2) - 3|) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent...
6
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = (-1(5((-2((-2x + 3)/(3x - 1)) - 3)/(1((-2x + 3)/(3x - 1)) - 3)) + 3) + 2)/(-1(5((-2((-2x + 3)/(3x - 1)) - 3)/(1((-2x + 3)/(3x - 1)) - 3)) + 3) - 1) and g(x) = 1|(2|(2|x| - 1) - 1| + 3)| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the...
5
MATH
[ { "content": "Find the number of times the graphs of f(x) = (2(2|(-5(3x - 3)^2 - 4(3x - 3) + 3)|) - 3)/(-2(2|(-5(3x - 3)^2 - 4(3x - 3) + 3)|) + 3) and g(x) = 4(-1(2|x + 3| - 3)^2 + 5(2|x + 3| - 3) - 5) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "ro...
4
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -2(1cos(3π(2((1x)/(-2x + 3)) - 5) + 2) - 2)^2 + 4 and g(x) = -2cos(3π(3(1|x + 1| - 1) - 3) - 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
104
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -2cos(1π(-1|(-1|(-4x)|)| - 1)) - 2 and g(x) = -1(3sin(3π((1x)/(-3x - 1))) - 3)^2 - 3(3sin(3π((1x)/(-3x - 1))) - 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ...
26
MATH
[ { "content": "Find the number of times the graphs of f(x) = -5(4(-1sin(1π(-1|x + 1|) + 3) - 2) + 5)^2 + 2(4(-1sin(1π(-1|x + 1|) + 3) - 2) + 5) + 5 and g(x) = 4(-2(3cos(3πx + 3) - 1) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }...
18
MATH
[ { "content": "How many solutions does the equation (2(1(-5(2|x - 3| - 3)^2 + 1(2|x - 3| - 3) + 3)) - 2)/(-2(1(-5(2|x - 3| - 3)^2 + 1(2|x - 3| - 3) + 3)) - 2) = -2cos(3π((-2(5x^2 - 1x - 3) - 1)/(-2(5x^2 - 1x - 3) - 1)) - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{...
4
MATH
[ { "content": "Find the number of times the graphs of f(x) = -5(-3sin(3π(3|(-2x^2 + 4x - 5) - 1| + 2) + 1))^2 + 5(-3sin(3π(3|(-2x^2 + 4x - 5) - 1| + 2) + 1)) - 1 and g(x) = (1((-1((1x - 1)/(-1x + 3)) - 3)/(1((1x - 1)/(-1x + 3)) + 1)) + 1)/(-1((-1((1x - 1)/(-1x + 3)) - 3)/(1((1x - 1)/(-1x + 3)) + 1)) - 3) interse...
600
MATH
[ { "content": "How many solutions does the equation 5(3|(1sin(3π(-1x - 2))) - 1| - 2)^2 - 3 = -2e^(2(-1|(-1x + 2) + 2| + 2)) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
100
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = 1|(-3((-2(2sin(1πx + 2) - 2) + 2)/(3(2sin(1πx + 2) - 2) + 2)) + 3) - 1| + 1 and g(x) = -5(5((2x + 2)/(-2x - 3))^2 - 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" ...
4
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = (2(2e^(1(3|(3|x - 3|) - 1| + 2))) + 2)/(1(2e^(1(3|(3|x - 3|) - 1| + 2))) - 2) and g(x) = -3((-1(-2sin(3πx + 2)))/(2(-2sin(3πx + 2)) - 1)) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", ...
60
MATH
[ { "content": "Find the number of times the graphs of f(x) = -3|(-2(3|(2cos(1πx + 2)) + 1| + 2)^2 + 2(3|(2cos(1πx + 2)) + 1| + 2) - 5) + 1| - 3 and g(x) = -4(1sin(2π(-2x^2 + 1x - 3) - 1) - 3)^2 + 1(1sin(2π(-2x^2 + 1x - 3) - 1) - 3) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \...
140
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 1|(-2(2(4x - 5)^2 + 4(4x - 5) - 3) - 3) + 1| - 2 and g(x) = 3cos(3π(1(-2|x + 2| - 3)^2 + 1(-2|x + 2| - 3) - 3) - 3) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ...
48
MATH
[ { "content": "Find the number of times the graphs of f(x) = -3((-1(-1|(-2sin(1πx - 2) - 3)| + 2) + 3)/(-2(-1|(-2sin(1πx - 2) - 3)| + 2) - 3)) - 5 and g(x) = (-3(2|(2x^2 + 4x + 1) + 1| + 1) + 2)/(-2(2|(2x^2 + 4x + 1) + 1| + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\box...
20
MATH
[ { "content": "How many solutions does the equation (-2(1|(-2|(4x^2 - 5) - 2|)| + 2) + 2)/(-1(1|(-2|(4x^2 - 5) - 2|)| + 2) + 3) = 1cos(2π(-2|(-1|x - 2| - 3) - 2| + 3) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
80
MATH
[ { "content": "Find the number of times the graphs of f(x) = -2|((2((3(2sin(3πx) - 3) - 3)/(-2(2sin(3πx) - 3) + 3)) + 1)/(-3((3(2sin(3πx) - 3) - 3)/(-2(2sin(3πx) - 3) + 3)) + 1)) + 2| - 1 and g(x) = (-2(-3sin(1π(1|x| - 3) - 3) + 1) - 3)/(-3(-3sin(1π(1|x| - 3) - 3) + 1) - 1) intersect in the interval [-10, 10].\n...
20
MATH
[ { "content": "How many points of intersection exist between the functions f(x) = -2|((3(-2(1x^2 + 4x - 3)^2 - 3(1x^2 + 4x - 3) + 1) + 3)/(3(-2(1x^2 + 4x - 3)^2 - 3(1x^2 + 4x - 3) + 1))) + 1| - 2 and g(x) = (2((3(1|x + 3| + 3))/(-1(1|x + 3| + 3) - 2)))/(-1((3(1|x + 3| + 3))/(-1(1|x + 3| + 3) - 2)) - 2) in the ra...
5
MATH
[ { "content": "How many solutions does the equation 1|(-1(-3sin(1π(4x^2 - 5x - 2) - 1))^2 - 1(-3sin(1π(4x^2 - 5x - 2) - 1)) - 2) + 3| + 2 = 3|(-2cos(1π((-2x + 1)/(-1x + 2)) + 2) - 2) - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
368
MATH
[ { "content": "Determine the number of intersection points between the functions f(x) = 2|((-2(2cos(3π(3x + 1) - 3)) + 3)/(2(2cos(3π(3x + 1) - 3)) - 1)) + 2| - 3 and g(x) = 3|(1e^(-1(-3|x| + 1)) + 3) + 2| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }...
20
MATH
[ { "content": "How many solutions does the equation -1|(-1|((1(2x^2 - 3x - 2))/(-1(2x^2 - 3x - 2) - 2)) - 3|) + 2| + 2 = (1(1(1|x - 3| + 1) - 1) - 3)/(1(1(1|x - 3| + 1) - 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
4
MATH
[ { "content": "How many solutions does the equation -5(-2(2cos(2π(1x + 2) - 2) - 3)^2 + 5(2cos(2π(1x + 2) - 2) - 3) + 1)^2 + 1(-2(2cos(2π(1x + 2) - 2) - 3)^2 + 5(2cos(2π(1x + 2) - 2) - 3) + 1) - 3 = -3(-3(-1x^2 + 2x + 5) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxe...
20
MATH
[ { "content": "Find the number of times the graphs of f(x) = 1e^(-1((3(-3(-4x^2 - 5x + 3) - 3) + 3)/(1(-3(-4x^2 - 5x + 3) - 3) - 3))) - 2 and g(x) = 2(-2(1cos(3πx + 2) + 1)^2 - 2(1cos(3πx + 2) + 1) + 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "ro...
60
MATH