messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(1(-2|(2x - 5) + 2| + 3) + 4)^2 - 2(1(-2|(2x - 5) + 2| + 3) + 4) - 2 and g(x) = -5(-2cos(1π(-2x - 5) - 2) - 2) - 3 where -10 ≤ x ≤ 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(-2sin(1π((-2x + 3)/(2x + 1)) - 2) + 3)^2 - 1(-2sin(1π((-2x + 3)/(2x + 1)) - 2) + 3) - 5) + 1| - 2 and g(x) = (2(-2sin(3π(-3|x + 1|) - 3) - 3) + 2)/(1(-2sin(3π(-3|x + 1|) - 3) - 3) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent t... | 55 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-2|(-2|(-2sin(1πx + 1) - 3) + 2| + 2) + 1| - 2)| - 3 and g(x) = -2sin(2π((2((-1x + 3)/(3x)) - 1)/(-1((-1x + 3)/(3x)) - 2)) - 2) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 82 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(3(-2|(1x - 5) + 1| + 1) + 1)^2 - 2(3(-2|(1x - 5) + 1| + 1) + 1) - 1 and g(x) = 2e^(1(2(-2x + 1) + 5)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(1((-2(3|x + 2| - 1) - 1)/(2(3|x + 2| - 1) - 1))^2 + 1((-2(3|x + 2| - 1) - 1)/(2(3|x + 2| - 1) - 1))) - 1)/(1(1((-2(3|x + 2| - 1) - 1)/(2(3|x + 2| - 1) - 1))^2 + 1((-2(3|x + 2| - 1) - 1)/(2(3|x + 2| - 1) - 1))) + 3) and g(x... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(2π(3((-1(-1x + 4) + 1)/(1(-1x + 4) + 3)) + 5) - 3) - 1 and g(x) = (-2(2e^(-1(-4x^2 + 1x + 3)) + 2) - 3)/(-2(2e^(-1(-4x^2 + 1x + 3)) + 2) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 154 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(2|(-3cos(2π(2x^2 - 3x - 5) + 1) + 1) - 1| + 2)^2 + 3(2|(-3cos(2π(2x^2 - 3x - 5) + 1) + 1) - 1| + 2) - 3 and g(x) = -2((-2(4x^2 - 2x - 4) - 3)/(-1(4x^2 - 2x - 4)))^2 + 4((-2(4x^2 - 2x - 4) - 3)/(-1(4x^2 - 2x - 4))) - 1 wher... | 7 | MATH |
[
{
"content": "How many solutions does the equation -1sin(3π(-2(-5(-3x^2 + 4)^2 - 4(-3x^2 + 4) + 1)^2 + 4(-5(-3x^2 + 4)^2 - 4(-3x^2 + 4) + 1) + 4) + 1) - 3 = 2sin(3π(-2|(-1x - 1)| + 2) + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 102 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-4(3(3sin(2πx - 3)) - 5)^2 + 4(3(3sin(2πx - 3)) - 5) + 4)^2 - 2(-4(3(3sin(2πx - 3)) - 5)^2 + 4(3(3sin(2πx - 3)) - 5) + 4) - 1 and g(x) = -3sin(1π(1|((1x + 2)/(-1x - 1)) + 1| - 2) - 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent th... | 39 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((-3((-2(1|x - 2| - 3) - 2)/(1(1|x - 2| - 3))) - 3)/(-1((-2(1|x - 2| - 3) - 2)/(1(1|x - 2| - 3))) + 3)) - 3| + 2 and g(x) = 3cos(1π(4((-2x + 3)/(2x - 3))^2 + 5((-2x + 3)/(2x - 3)) - 1) - 2) - 1 where -10 ≤ x ≤ 10.\n\nPrese... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1((1((1x - 1)/(1x + 1)) - 2)/(-3((1x - 1)/(1x + 1)) - 3)) + 4) + 2)/(2(-1((1((1x - 1)/(1x + 1)) - 2)/(-3((1x - 1)/(1x + 1)) - 3)) + 4) - 1) and g(x) = 1|(-1|(5x^2 + 1x - 2) + 1| + 3) + 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answ... | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(1(-2|(1x + 1) - 3| + 2)^2 - 5)| + 3 and g(x) = -3cos(2π((-1((-3x - 1)/(3x + 2)) - 3)/(3((-3x - 1)/(3x + 2)) - 2)) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(1π(-2|(-5(-2|x + 1| + 2)) + 1| + 3) + 3) + 2 and g(x) = 4((-1(-3sin(3πx + 2) + 2))/(1(-3sin(3πx + 2) + 2) - 3)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 135 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(1π((-2(-1(-1|x + 3| + 1)) - 1)/(-3(-1(-1|x + 3| + 1)) + 1)) + 3) + 3 and g(x) = 3(2|(2e^(1x) - 1) - 2| - 2) - 5 intersect 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 (1(2e^(-1(3((-2x + 3)/(-1x))^2 + 5((-2x + 3)/(-1x)) - 5))) + 1)/(3(2e^(-1(3((-2x + 3)/(-1x))^2 + 5((-2x + 3)/(-1x)) - 5))) - 2) = 3sin(2π((1(-1x^2 - 2x + 1) + 1)/(-3(-1x^2 - 2x + 1) - 1)) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in L... | 29 | MATH |
[
{
"content": "How many solutions does the equation 2((-3(-1(-2x^2 - 2x + 2)^2 + 2) - 3)/(1(-1(-2x^2 - 2x + 2)^2 + 2) - 2)) + 1 = 2|(-3sin(1π(-1x^2 + 4x + 3) - 1) - 2) + 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-2(-2cos(2π(-2|x| - 2) + 3) - 3) - 4) + 4 and g(x) = -3e^(1(-3|(3x) + 2| - 2)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many solutions does the equation (-3((-2(-2e^(1(-2x^2 + 4)) + 1) + 1)/(-3(-2e^(1(-2x^2 + 4)) + 1) - 3)) + 2)/(-3((-2(-2e^(1(-2x^2 + 4)) + 1) + 1)/(-3(-2e^(1(-2x^2 + 4)) + 1) - 3)) - 2) = 2|(1(-1|x + 2| + 3)^2 + 5(-1|x + 2| + 3) - 1)| - 3 have for x in the interval [-10, 10]?\n\nPresent the ans... | 9 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(1π(3(-2(2x) + 4) + 4) + 3) - 2 and g(x) = 2sin(3π((2(1x^2 + 4) + 2)/(1(1x^2 + 4) + 3)) - 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 237 | MATH |
[
{
"content": "How many solutions does the equation (1(4((-1(3cos(1πx - 3) - 1) - 1)/(-3(3cos(1πx - 3) - 1)))^2 - 3((-1(3cos(1πx - 3) - 1) - 1)/(-3(3cos(1πx - 3) - 1))) + 4))/(1(4((-1(3cos(1πx - 3) - 1) - 1)/(-3(3cos(1πx - 3) - 1)))^2 - 3((-1(3cos(1πx - 3) - 1) - 1)/(-3(3cos(1πx - 3) - 1))) + 4)) = (-1(3sin(2π(1... | 40 | MATH |
[
{
"content": "How many solutions does the equation 2(2|(3((3x - 2)/(-1x - 3)) - 1) - 2| - 3) - 5 = (2(-2sin(1π((-3x - 1)/(-1x + 3))) + 2) - 2)/(2(-2sin(1π((-3x - 1)/(-1x + 3))) + 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(3π(-5(4(4x^2 - 1x + 1) - 3)^2 + 3(4(4x^2 - 1x + 1) - 3)) + 2) + 3 and g(x) = -2sin(2π(-4((-1x + 1)/(-1x + 3)) + 4) - 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 121 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(3((-2(1|x + 3| - 2))/(1(1|x + 3| - 2) + 1)) + 5) + 2 and g(x) = -3cos(1π(3(1x + 1) + 1) + 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(3(-2((-3x + 3)/(3x + 3))^2 - 5((-3x + 3)/(3x + 3)) - 3)) - 3 and g(x) = (-1(-1|((3x)/(-1x - 1)) + 3| + 2) + 3)/(2(-1|((3x)/(-1x - 1)) + 3| + 2) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-3|(1|(2x + 5)| - 2) - 3| + 1) - 5 and g(x) = (-1(-1(-4x^2 - 5x - 3)^2 - 3(-4x^2 - 5x - 3) - 2) + 2)/(-1(-1(-4x^2 - 5x - 3)^2 - 3(-4x^2 - 5x - 3) - 2)) intersect 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) = 2(4((2(-2cos(3πx + 3) + 3))/(-3(-2cos(3πx + 3) + 3) + 2))^2 + 4((2(-2cos(3πx + 3) + 3))/(-3(-2cos(3πx + 3) + 3) + 2)) + 5)^2 - 4(4((2(-2cos(3πx + 3) + 3))/(-3(-2cos(3πx + 3) + 3) + 2))^2 + 4((2(-2cos(3πx + 3) + 3))/(-3(-2cos(... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3(5((-3x + 3)/(-3x + 2)) + 5) - 5) + 1 and g(x) = 1((1(3sin(1πx - 1) - 3) - 3)/(1(3sin(1πx - 1) - 3) + 3)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(5(-2|(-5x^2 + 5x - 2) + 1|) - 2) + 4 and g(x) = 1sin(3π(4(-3|x - 2| - 3) - 5) + 1) - 2 in the range -10 ≤ x ≤ 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) = (-1((2(-5(-1x + 3)^2 - 3(-1x + 3) + 3) - 2)/(-1(-5(-1x + 3)^2 - 3(-1x + 3) + 3) + 2)) - 3)/(-1((2(-5(-1x + 3)^2 - 3(-1x + 3) + 3) - 2)/(-1(-5(-1x + 3)^2 - 3(-1x + 3) + 3) + 2)) + 2) and g(x) = 2|(-3e^(-1(4x - 2)) + 2)| - 3 intersect in the interval [-1... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2|(-4(1|(5x^2 - 5x - 3) + 2| - 3) - 4) + 2| - 1 = -5((-1(3sin(3πx - 3)) - 2)/(1(3sin(3πx - 3)) + 3))^2 + 1((-1(3sin(3πx - 3)) - 2)/(1(3sin(3πx - 3)) + 3)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-2(2cos(2π(-1|x| + 2) + 3) + 1) + 5) - 3| - 1 and g(x) = (3(1cos(2π(2|x + 1| + 2) + 2) - 2))/(3(1cos(2π(2|x + 1| + 2) + 2) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1(2cos(2π(-3x - 4) + 3))^2 - 3(2cos(2π(-3x - 4) + 3)) + 1)^2 - 4(1(2cos(2π(-3x - 4) + 3))^2 - 3(2cos(2π(-3x - 4) + 3)) + 1) - 3 and g(x) = -5((3(3x^2 - 4x + 3) + 2)/(2(3x^2 - 4x + 3) - 1)) + 4 in the range -10 ≤ x ≤ 10?\n\nPrese... | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-1sin(3π(-4(-1|x - 1| - 1) + 5) - 2) - 3) + 1| + 1 and g(x) = (-3(-3(-3cos(3πx - 2) + 1) - 2) - 2)/(-1(-3(-3cos(3πx - 2) + 1) - 2) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1e^(-2(1|((-2(-2x + 4) - 2)/(-3(-2x + 4) + 1)) + 2|)) + 3 and g(x) = -1(2(2|x|)^2 - 4(2|x|) - 5)^2 + 4(2(2|x|)^2 - 4(2|x|) - 5) + 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|(-1|(2|(2e^(-2x) - 2) - 3| - 3) + 3| + 1) - 2| + 3 and g(x) = -3(-4(2cos(2πx - 1) + 2) + 3) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-4(-1(-3|x + 3| - 3)^2 - 4(-3|x + 3| - 3) - 1) - 5))/(-3(-4(-1(-3|x + 3| - 3)^2 - 4(-3|x + 3| - 3) - 1) - 5)) and g(x) = (-1(5(3cos(1πx + 3) + 2)^2 + 3(3cos(1πx + 3) + 2) + 4) - 3)/(-2(5(3cos(1πx + 3) + 2)^2 + 3(3cos(1πx ... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((2((-2(-1x + 3))/(2(-1x + 3) + 1)) - 2)/(-1((-2(-1x + 3))/(2(-1x + 3) + 1)) + 2)) and g(x) = 1e^(-1(-2|(2|x - 2| - 2)| - 1)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 5 | MATH |
[
{
"content": "How many solutions does the equation -1cos(2π(3((1(4x^2 - 4x - 4))/(-2(4x^2 - 4x - 4) - 2)) - 3) - 3) + 1 = 2sin(2π(3|((2x + 2)/(-3x + 4))| - 1) + 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 114 | MATH |
[
{
"content": "How many solutions does the equation 1|(-1(-2sin(3π(-2x^2 + 2x + 5) - 1) + 2)) - 3| - 1 = -4(4(3cos(1πx + 2) + 1) + 3)^2 - 3(4(3cos(1πx + 2) + 1) + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(1|((2(-2|x - 2| - 1) - 3)/(1(-2|x - 2| - 1) + 2)) - 1| + 3) + 2)/(-3(1|((2(-2|x - 2| - 1) - 3)/(1(-2|x - 2| - 1) + 2)) - 1| + 3)) and g(x) = -2(1|(-1x^2 + 1x + 1) + 3| - 3) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(2cos(2π(2(2|x + 2| - 3)^2 + 5(2|x + 2| - 3) - 2) - 3) + 2) - 1)/(-2(2cos(2π(2(2|x + 2| - 3)^2 + 5(2|x + 2| - 3) - 2) - 3) + 2) + 3) and g(x) = 3e^(-2((3((-3x + 2)/(1x + 1)) + 3)/(2((-3x + 2)/(1x + 1)) - 1))) intersect in the interval [-10, 10].\n\n... | 39 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-3|(1(-4x^2 + 5x + 3) + 5) + 1|) - 5 and g(x) = (-2(4((1x - 2)/(1x + 1))^2 - 4((1x - 2)/(1x + 1)) - 2) - 1)/(-1(4((1x - 2)/(1x + 1))^2 - 4((1x - 2)/(1x + 1)) - 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\box... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-4(-4(3x^2 - 3x + 1) + 5) - 5)^2 - 2(-4(-4(3x^2 - 3x + 1) + 5) - 5) and g(x) = -3(-2|(-3x^2 + 4x + 4) + 3| + 2) - 2 where -10 ≤ x ≤ 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(-1((-1(-1|x + 3|) + 1)/(2(-1|x + 3|) - 1))^2 + 1((-1(-1|x + 3|) + 1)/(2(-1|x + 3|) - 1)) - 2) - 1 and g(x) = 1|(2sin(3π(2x^2 - 4))) + 1| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 762 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(2(-2(-3sin(1πx - 3) + 3)^2 + 3(-3sin(1πx - 3) + 3) - 1) - 4)| and g(x) = 3((-1((2x - 1)/(-3x + 2)) - 3)/(1((2x - 1)/(-3x + 2)) + 2)) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 41 | MATH |
[
{
"content": "How many solutions does the equation 2(1|(2cos(2π(-2|x - 3| + 2) - 2) + 3)| - 3)^2 + 3(1|(2cos(2π(-2|x - 3| + 2) - 2) + 3)| - 3) + 4 = 3|(-1cos(1π(-1x^2 - 5x - 3) + 3) + 2) + 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 82 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(5(1|(1x - 1) - 3| - 3) + 1) + 3)/(-2(5(1|(1x - 1) - 3| - 3) + 1) + 1) and g(x) = (-2(-5(1x + 3)^2 + 5(1x + 3) + 1) - 1)/(1(-5(1x + 3)^2 + 5(1x + 3) + 1)) intersect in the interval [-10, 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) = 2sin(2π(3((1(2|x + 3| + 3))/(-1(2|x + 3| + 3) + 2))^2 - 3((1(2|x + 3| + 3))/(-1(2|x + 3| + 3) + 2)) - 2) + 2) - 1 and g(x) = -2|(-1((2x + 2)/(1x - 2))^2 - 2((2x + 2)/(1x - 2)) + 5)| in the range -10 ≤ x ≤ 10?\n\nPresent the answer ... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(3|(-3sin(1π(4x - 4) + 2) - 2) + 1| + 1) + 2)/(-1(3|(-3sin(1π(4x - 4) + 2) - 2) + 1| + 1) + 3) and g(x) = 1|(1|(4x^2 + 4x + 3) - 1| + 2) + 3| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 66 | MATH |
[
{
"content": "How many solutions does the equation 1e^(-2(-3|((1(-3x) + 1)/(1(-3x))) - 1| + 2)) - 1 = 1e^(-2(2|(-5x^2 - 3x + 4) + 2|)) - 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 points of intersection exist between the functions f(x) = -3(1|(-4((-1x - 1)/(3x + 1)) + 1) - 3| + 1) - 2 and g(x) = -5(-3sin(3π(-1x^2 + 3x + 4) + 2) + 3)^2 - 5(-3sin(3π(-1x^2 + 3x + 4) + 2) + 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 584 | MATH |
[
{
"content": "How many solutions does the equation (3(-2sin(3π(3(-1|x| + 3) - 3) - 3) + 2))/(-1(-2sin(3π(3(-1|x| + 3) - 3) - 3) + 2) + 2) = -5(3|(3|x - 1| + 1)| - 2)^2 + 3(3|(3|x - 1| + 1)| - 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 6 | MATH |
[
{
"content": "How many solutions does the equation (-1((3(-2(-3|x + 1| + 1)^2 - 3(-3|x + 1| + 1) + 5) + 2)/(2(-2(-3|x + 1| + 1)^2 - 3(-3|x + 1| + 1) + 5) + 2)))/(-2((3(-2(-3|x + 1| + 1)^2 - 3(-3|x + 1| + 1) + 5) + 2)/(2(-2(-3|x + 1| + 1)^2 - 3(-3|x + 1| + 1) + 5) + 2)) + 3) = (-3(-3(-3e^(-1x) - 2) - 5) + 2)/(2(... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-1(-2|(-1x^2 - 2x + 3) + 2| + 1)^2 - 1(-2|(-1x^2 - 2x + 3) + 2| + 1)) + 2)/(3(-1(-2|(-1x^2 - 2x + 3) + 2| + 1)^2 - 1(-2|(-1x^2 - 2x + 3) + 2| + 1)) - 2) and g(x) = (-3(1(-3x^2 + 5x + 3)^2 - 5(-3x^2 + 5x + 3) + 1) - 1)/(1(1(-3x^2 + 5x + 3)^2 - 5(-3x... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(-4(4(2sin(2πx - 1))^2 + 4(2sin(2πx - 1)) - 2)^2 - 2(4(2sin(2πx - 1))^2 + 4(2sin(2πx - 1)) - 2) + 1) - 5 and g(x) = -2((-3(-2|x - 3| + 1) - 3)/(-3(-2|x - 3| + 1) + 4))^2 + 3((-3(-2|x - 3| + 1) - 3)/(-3(-2|x - 3| + 1) + 4)) + 1 in ... | 17 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|((-3(3|(5x^2 + 2x - 4) - 1| + 1) - 2)/(1(3|(5x^2 + 2x - 4) - 1| + 1) - 3)) + 1| - 3 and g(x) = -1(4(-2x^2 + 3x - 3) - 1)^2 - 4(4(-2x^2 + 3x - 3) - 1) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(3cos(3π(-5(1x - 2) - 4) - 1) + 1)^2 + 5(3cos(3π(-5(1x - 2) - 4) - 1) + 1) + 5 and g(x) = (-2(3sin(3π(-2|x - 1| - 1) + 3) + 2) - 1)/(3(3sin(3π(-2|x - 1| - 1) + 3) + 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\... | 17 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(2|(-1((1x + 2)/(1x + 3))^2 + 1((1x + 2)/(1x + 3)) + 5) + 3| - 3) + 3 and g(x) = 3cos(2π(1|(-2x - 1) + 3| - 3) + 3) - 2 where -10 ≤ x ≤ 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) = 1(-1(-1|(-3cos(2πx - 1) - 1)| - 3)^2 + 4(-1|(-3cos(2πx - 1) - 1)| - 3) - 3) - 5 and g(x) = 5((2(1x + 5) - 2)/(-1(1x + 5) - 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(2π(1(-2(1x^2 - 5x + 3)^2 + 2(1x^2 - 5x + 3) - 1) - 5) - 3) - 2 and g(x) = -5(-1(-4x + 2) + 5)^2 - 4(-1(-4x + 2) + 5) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-2cos(2π(3((2x + 3)/(-1x + 2)) + 4) - 3)) + 2| + 1 and g(x) = -3cos(1π((-3((-2x + 3)/(-1x + 3)) + 3)/(-2((-2x + 3)/(-1x + 3)) + 1)) - 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 217 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-1|((2((2x - 1)/(1x - 2)) + 3)/(2((2x - 1)/(1x - 2)) - 1))| + 2) - 2| + 1 and g(x) = (-3(-5(-1x + 3)^2 - 2(-1x + 3) + 5) + 2)/(-3(-5(-1x + 3)^2 - 2(-1x + 3) + 5)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Yo... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(5((-2(1x^2 - 5x + 3) + 3)/(-1(1x^2 - 5x + 3) - 2))^2 + 1((-2(1x^2 - 5x + 3) + 3)/(-1(1x^2 - 5x + 3) - 2)) - 2)^2 + 4(5((-2(1x^2 - 5x + 3) + 3)/(-1(1x^2 - 5x + 3) - 2))^2 + 1((-2(1x^2 - 5x + 3) + 3)/(-1(1x^2 - 5x + 3) - 2)) - 2) -... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-4(-1(3cos(1πx + 1) + 3) + 2) - 1) + 5 and g(x) = -1sin(1π(-2|((-1x + 2)/(1x - 2)) + 2| - 1) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(3e^(2(-1(2x^2 - 3x - 5)^2 + 3(2x^2 - 3x - 5) + 3))) + 3| + 2 and g(x) = 1|(-2((3x + 1)/(-2x - 3))^2 + 5((3x + 1)/(-2x - 3)) - 3) - 2| - 1 where -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) = -2sin(2π(4((-2(3x^2 + 4x - 4))/(1(3x^2 + 4x - 4) - 2)) + 1) - 1) - 2 and g(x) = 3cos(2π(2|((1x - 2)/(2x)) + 2| + 3) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 116 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2e^(2(-2(-2((-1x - 2)/(-2x - 2)) + 5)^2 - 5(-2((-1x - 2)/(-2x - 2)) + 5) - 3)) + 1 and g(x) = 4(-5(-3cos(2πx + 3) + 2) + 1)^2 - 1(-5(-3cos(2πx + 3) + 2) + 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-1(-3((-2x - 1)/(2x - 1)) + 4)^2 - 4(-3((-2x - 1)/(2x - 1)) + 4) - 3) + 1| - 2 and g(x) = 3(-1|(-5x^2 + 1x - 1) - 1| + 3)^2 - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-2|((1(-1x^2 - 3x - 1) - 1)/(3(-1x^2 - 3x - 1))) - 1| + 3)^2 + 2(-2|((1(-1x^2 - 3x - 1) - 1)/(3(-1x^2 - 3x - 1))) - 1| + 3) + 5 and g(x) = (1((-2((-3x)/(-3x - 2)) + 3)/(-2((-3x)/(-3x - 2)) + 1)) + 1)/(-3((-2((-3x)/(-3x - 2)) + 3... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2e^(2(-2|(-1((2x - 2)/(3x))^2 - 4((2x - 2)/(3x)))| + 2)) and g(x) = 1|(1((3x - 3)/(-2x + 3)) - 5) + 1| + 2 intersect 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 (-2((1(-2(-2sin(1πx + 2) + 1) + 1) + 2)/(2(-2(-2sin(1πx + 2) + 1) + 1) - 3)) + 3)/(-3((1(-2(-2sin(1πx + 2) + 1) + 1) + 2)/(2(-2(-2sin(1πx + 2) + 1) + 1) - 3)) + 4) = -4(-2|(1|x + 2| - 2)|) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-1|((1(2x^2 + 4x) - 2)/(1(2x^2 + 4x) - 3)) - 1| + 2)^2 - 4(-1|((1(2x^2 + 4x) - 2)/(1(2x^2 + 4x) - 3)) - 1| + 2) - 4 and g(x) = 2(-2|(-3|x + 1| + 1) + 3| + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxe... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3e^(-2(2((2(3x^2 + 5x - 5) - 1)/(-2(3x^2 + 5x - 5) - 1))^2 + 4((2(3x^2 + 5x - 5) - 1)/(-2(3x^2 + 5x - 5) - 1)) + 1)) + 2 and g(x) = (2(2e^(1(1|x + 3|)) - 2) - 2)/(-1(2e^(1(1|x + 3|)) - 2) - 2) where -10 ≤ x ≤ 10.\n\nPresent ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1e^(1((-2(1|(2x^2 - 1) + 2| + 1) - 1)/(-2(1|(2x^2 - 1) + 2| + 1) + 3))) + 1 and g(x) = (-1(3cos(1π(5x + 4) + 3) + 2) - 2)/(1(3cos(1π(5x + 4) + 3) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 96 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((1(1(1|x - 1| - 3) + 3) - 3)/(1(1(1|x - 1| - 3) + 3) + 3)) + 1| and g(x) = 3(1cos(2π(5x + 4) + 3) + 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 180 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(1π(-2|(-3(1|x| + 2)^2 - 5(1|x| + 2) - 4) + 1| + 2) - 1) + 3 and g(x) = 1|(-5(-5x + 5)^2 - 1) + 3| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many solutions does the equation -1|((1(2e^(-1(-3|x| + 2))) + 1)/(2(2e^(-1(-3|x| + 2))))) + 1| = 1(-4(2|x + 2| - 1)^2 + 3(2|x + 2| - 1) + 4)^2 - 3(-4(2|x + 2| - 1)^2 + 3(2|x + 2| - 1) + 4) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(3((-3(3x^2 - 5x))/(2(3x^2 - 5x) - 3))^2 - 4((-3(3x^2 - 5x))/(2(3x^2 - 5x) - 3)) - 3) + 4 and g(x) = 4(4(3cos(2πx) + 2) - 5) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 38 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(1π(-4((2(2|x + 2|))/(2(2|x + 2|) + 1))^2 - 1((2(2|x + 2|))/(2(2|x + 2|) + 1)) - 4) + 2) + 1 and g(x) = (3(-1|(-4x - 5)| + 3) - 3)/(3(-1|(-4x - 5)| + 3) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\b... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-2(2|(-1|x - 3| + 1) + 1| - 2)^2 - 2(2|(-1|x - 3| + 1) + 1| - 2) + 5) + 1)/(-1(-2(2|(-1|x - 3| + 1) + 1| - 2)^2 - 2(2|(-1|x - 3| + 1) + 1| - 2) + 5) + 2) and g(x) = (2(-3|(-1x^2 - 2x + 4)| + 3) - 3)/(3(-3|(-1x^2 - 2x + 4)| + 3) - 2) intersect in the... | 8 | MATH |
[
{
"content": "How many solutions does the equation (2(4(-2|(-5x - 2) + 2| + 3)^2 + 5(-2|(-5x - 2) + 2| + 3) - 4) + 1)/(2(4(-2|(-5x - 2) + 2| + 3)^2 + 5(-2|(-5x - 2) + 2| + 3) - 4) - 1) = 1cos(2π((-3(5x^2 + 1x + 2) - 2)/(-1(5x^2 + 1x + 2))) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(3sin(1π(2|((3x - 1)/(-1x - 1)) - 1| - 1) + 3) + 2) + 1 and g(x) = -5(3|((3x - 3)/(-3x - 1)) + 2| - 2)^2 + 1(3|((3x - 3)/(-3x - 1)) + 2| - 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(5(2cos(2π(3|x + 1| - 1) + 1) + 2)^2 - 5(2cos(2π(3|x + 1| - 1) + 1) + 2) + 5) + 3)/(2(5(2cos(2π(3|x + 1| - 1) + 1) + 2)^2 - 5(2cos(2π(3|x + 1| - 1) + 1) + 2) + 5) + 1) and g(x) = -2(2sin(3π(2|x + 2| - 3) - 1)) + 2 intersect in the interval [-10, 10]... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-2cos(2π((-3(-3|x + 2|))/(-2(-3|x + 2|) - 2)) - 3) - 2)^2 + 3(-2cos(2π((-3(-3|x + 2|))/(-2(-3|x + 2|) - 2)) - 3) - 2) - 2 and g(x) = -2e^(1(5(5x - 3) - 1)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your a... | 61 | MATH |
[
{
"content": "How many solutions does the equation 1(-2(3cos(2π(3|x - 1| - 3) + 2))^2 - 4(3cos(2π(3|x - 1| - 3) + 2)) + 1)^2 + 4(-2(3cos(2π(3|x - 1| - 3) + 2))^2 - 4(3cos(2π(3|x - 1| - 3) + 2)) + 1) + 1 = -1(-2(-5x - 2) - 5)^2 - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-3sin(3π(4(1x - 1)^2 + 2(1x - 1) - 5) + 1) - 1) - 3| + 2 and g(x) = -1(-4((-3x - 1)/(2x + 2))^2 + 2((-3x - 1)/(2x + 2)) + 1)^2 - 3(-4((-3x - 1)/(2x + 2))^2 + 2((-3x - 1)/(2x + 2)) + 1) + 4 where -10 ≤ x ≤ 10.\n\nPresent t... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-2(-1(-1sin(2πx + 3) - 1) - 4) + 1)/(1(-1(-1sin(2πx + 3) - 1) - 4) + 3))^2 - 5((-2(-1(-1sin(2πx + 3) - 1) - 4) + 1)/(1(-1(-1sin(2πx + 3) - 1) - 4) + 3)) - 4 and g(x) = (1(-2(-3x^2 - 4x - 2)^2 + 1(-3x^2 - 4x - 2) + 3) + 3)/(-2(-2(-3x^2 - 4x - 2)^2 + ... | 42 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-3(-2|((-3x - 2)/(-3x - 3))| - 1) - 2)/(1(-2|((-3x - 2)/(-3x - 3))| - 1)))^2 + 2((-3(-2|((-3x - 2)/(-3x - 3))| - 1) - 2)/(1(-2|((-3x - 2)/(-3x - 3))| - 1))) + 1 and g(x) = 1sin(1π((2(2|x - 1| - 3) - 2)/(-2(2|x - 1| - 3) + ... | 29 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(1π((1(-2|((-1x - 1)/(2x - 3))| + 3) - 3)/(2(-2|((-1x - 1)/(2x - 3))| + 3) - 3)) - 1) + 2 and g(x) = 1sin(1π(-2(-2x + 4)^2 - 1(-2x + 4) - 1)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 836 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-2sin(3π((-2((1x + 2)/(-2x + 3)) + 3)/(2((1x + 2)/(-2x + 3)) + 3)) + 2) + 2)^2 + 4(-2sin(3π((-2((1x + 2)/(-2x + 3)) + 3)/(2((1x + 2)/(-2x + 3)) + 3)) + 2) + 2) - 3 and g(x) = -3|(3|(-4x - 5) + 1| - 2)| - 3 intersect in the interval [-10, 10].\n\nPre... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|((3((1(3|x + 3| + 3) - 1)/(-1(3|x + 3| + 3))) - 3)/(3((1(3|x + 3| + 3) - 1)/(-1(3|x + 3| + 3))) + 3)) - 1| + 2 and g(x) = (-1(1|(1cos(3πx) + 1) + 3| - 1) + 1)/(1(1|(1cos(3πx) + 1) + 3| - 1) - 3) in the range -10 ≤ x ≤ 10?\n\nPres... | 56 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(3((-2(-1x^2 - 3x + 4))/(-3(-1x^2 - 3x + 4))) - 1))/(3(3((-2(-1x^2 - 3x + 4))/(-3(-1x^2 - 3x + 4))) - 1) - 2) and g(x) = (-2(-3cos(2π((-1x)/(-1x + 2)) - 1) + 3) + 1)/(3(-3cos(2π((-1x)/(-1x + 2)) - 1) + 3) - 2) where -10 ≤ x... | 13 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-1(2|(4x^2 + 3x + 3) + 1| - 1)) - 1)/(1(-1(2|(4x^2 + 3x + 3) + 1| - 1)) - 3) and g(x) = -3cos(3π(2|((1x - 2)/(2x + 1)) + 2|) - 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 100 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-2sin(1π(-1(1x^2 + 3x - 5) - 4) + 2) - 2) + 5 and g(x) = -3(2(-3|x + 2| + 3) + 5)^2 + 5(2(-3|x + 2| + 3) + 5) + 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 (2(-2|(2(-1cos(1πx + 3))^2 + 3(-1cos(1πx + 3)) + 1)| + 1) + 2)/(3(-2|(2(-1cos(1πx + 3))^2 + 3(-1cos(1πx + 3)) + 1)| + 1) - 1) = 2cos(3π((-2((2x)/(-3x + 2)) - 2)/(-3((2x)/(-3x + 2)) + 1)) - 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in La... | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3(3cos(3π(-2|x + 2| + 1) + 1))^2 + 2(3cos(3π(-2|x + 2| + 1) + 1)) + 1) + 1 and g(x) = (3(-3sin(1π(-1x - 3) - 2) - 3) - 2)/(1(-3sin(1π(-1x - 3) - 2) - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your an... | 240 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-4((3(1|x - 3| - 3) - 3)/(1(1|x - 3| - 3))) - 1) - 3)/(1(-4((3(1|x - 3| - 3) - 3)/(1(1|x - 3| - 3))) - 1)) and g(x) = 3sin(2π(1((-1x - 2)/(3x - 2))^2 + 3) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Y... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((1(-2sin(3π(4x^2 + 2x - 3) + 3) - 1) + 3)/(2(-2sin(3π(4x^2 + 2x - 3) + 3) - 1) - 1)) + 1| + 1 and g(x) = 4(-4(-1|x + 1| + 3) - 2) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 174 | MATH |
[
{
"content": "How many solutions does the equation 4((1(2sin(3π(3|x - 3| - 2) + 1) - 2) + 2)/(-1(2sin(3π(3|x - 3| - 2) + 1) - 2)))^2 + 5((1(2sin(3π(3|x - 3| - 2) + 1) - 2) + 2)/(-1(2sin(3π(3|x - 3| - 2) + 1) - 2))) - 2 = -1(1sin(2π(-5x^2 - 1x - 2) - 1) + 1)^2 + 1(1sin(2π(-5x^2 - 1x - 2) - 1) + 1) + 1 have for x... | 182 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((-3(-2|(-4x + 1)| + 3) - 2)/(-3(-2|(-4x + 1)| + 3)))^2 - 4((-3(-2|(-4x + 1)| + 3) - 2)/(-3(-2|(-4x + 1)| + 3))) + 2 and g(x) = (1((-2(1|x|) - 3)/(-1(1|x|) - 1)) - 1)/(-3((-2(1|x|) - 3)/(-1(1|x|) - 1)) + 1) in the range -10 ≤ x ≤... | 4 | MATH |
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