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) = -2|((1(2(-1((-1x - 1)/(3x - 2))^2 + 5((-1x - 1)/(3x - 2)) + 2) + 5) + 1)/(1(2(-1((-1x - 1)/(3x - 2))^2 + 5((-1x - 1)/(3x - 2)) + 2) + 5) - 3)) + 3| + 3 and g(x) = -2|(2|((1((-2x + 2)/(-2x - 2)) + 2)/(-3((-2x + 2)/(-2x - 2)) -... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-1(3e^(-1(-5(-1x - 1)^2 - 2(-1x - 1) + 5)) - 3) - 4) - 3| and g(x) = 2|(1(-3|(-4x - 4) - 1| + 2)^2 - 5(-3|(-4x - 4) - 1| + 2) - 5) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2e^(-1(5((-2(-2(-3x - 5)^2 + 5(-3x - 5)) + 2)/(-2(-2(-3x - 5)^2 + 5(-3x - 5)) - 1))^2 + 5((-2(-2(-3x - 5)^2 + 5(-3x - 5)) + 2)/(-2(-2(-3x - 5)^2 + 5(-3x - 5)) - 1)) - 2)) - 1 and g(x) = -1(-1cos(1π(-1|(-3x^2 + 4x) - 3| - 2) - 2) - ... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-5(-3|(-3|(1x^2 + 5x + 4) + 1| + 2) - 2| - 1)^2 - 3(-3|(-3|(1x^2 + 5x + 4) + 1| + 2) - 2| - 1) + 1) - 1)/(2(-5(-3|(-3|(1x^2 + 5x + 4) + 1| + 2) - 2| - 1)^2 - 3(-3|(-3|(1x^2 + 5x + 4) + 1| + 2) - 2| - 1) + 1) - 3) and g(x) ... | 300 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((-3((-3(5(1|x + 1|)^2 + 1(1|x + 1|) - 1) - 2)/(3(5(1|x + 1|)^2 + 1(1|x + 1|) - 1) + 1)))/(1((-3(5(1|x + 1|)^2 + 1(1|x + 1|) - 1) - 2)/(3(5(1|x + 1|)^2 + 1(1|x + 1|) - 1) + 1)) + 1)) + 2| + 1 and g(x) = 1(-3(-1cos(1π(-3x +... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(5(2|(-4((-2x - 2)/(-2x))) - 2| - 3) + 1) - 3| - 3 and g(x) = 1|(1(5(2cos(2πx - 3))^2 + 5(2cos(2πx - 3)) + 1)^2 - 5(5(2cos(2πx - 3))^2 + 5(2cos(2πx - 3)) + 1) - 1) - 2| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTe... | 42 | MATH |
[
{
"content": "How many solutions does the equation (-3(4(-3(-2|(-1x^2 + 3x + 2)| + 1) + 1) - 4))/(-2(4(-3(-2|(-1x^2 + 3x + 2)| + 1) + 1) - 4) - 2) = 4(3e^(-1(1(1x + 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": "Determine the number of intersection points between the functions f(x) = (1(1(-5(5((-2x + 1)/(2x))^2 - 4((-2x + 1)/(2x)))^2 + 4(5((-2x + 1)/(2x))^2 - 4((-2x + 1)/(2x))) + 5) + 3) + 3)/(-2(1(-5(5((-2x + 1)/(2x))^2 - 4((-2x + 1)/(2x)))^2 + 4(5((-2x + 1)/(2x))^2 - 4((-2x + 1)/(2x))) + 5) + 3) + 3) an... | 963 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(2(-2e^(2((-1(3|x + 1|) + 2)/(-1(3|x + 1|) - 3))) - 1) + 3) + 2)/(2(2(-2e^(2((-1(3|x + 1|) + 2)/(-1(3|x + 1|) - 3))) - 1) + 3) + 3) and g(x) = -3(-2|(-5((-3x + 1)/(3x)) - 5) + 1|) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTe... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π((-2(1(-3|(-2x + 1)|) - 2) + 3)/(-2(1(-3|(-2x + 1)|) - 2) - 3)) + 1) + 2 and g(x) = 1(-2sin(1π(1(4x^2) - 2) + 3) + 1)^2 - 1(-2sin(1π(1(4x^2) - 2) + 3) + 1) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex forma... | 21 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(3π(3((-3((-1(-2|x + 3| - 3) - 2)/(-3(-2|x + 3| - 3))) - 3)/(2((-1(-2|x + 3| - 3) - 2)/(-3(-2|x + 3| - 3)))))^2 + 2((-3((-1(-2|x + 3| - 3) - 2)/(-3(-2|x + 3| - 3))) - 3)/(2((-1(-2|x + 3| - 3) - 2)/(-3(-2|x + 3| - 3))))) - 3) + 3) + 2 and g(x) = 2|(... | 181 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-2|(2e^(1(-1(-1x + 2)^2 + 4(-1x + 2) - 5)) + 1) + 2| - 3) + 2 and g(x) = 4((-1(-3cos(3π(-4x^2 + 5x) - 1) + 1) + 2)/(-3(-3cos(3π(-4x^2 + 5x) - 1) + 1) + 2)) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\bo... | 41 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2e^(1(-2(-2|(-1(3x^2 + 5x - 2))| + 3)^2 - 1(-2|(-1(3x^2 + 5x - 2))| + 3) + 5)) and g(x) = 2cos(3π(3|(-1|(1x^2 + 4x + 1)| + 3) + 1| + 3) - 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 1359 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|((-3(-4((-2(-5x + 2) + 2)/(2(-5x + 2) - 1))^2 + 5((-2(-5x + 2) + 2)/(2(-5x + 2) - 1)) - 4) - 2)/(3(-4((-2(-5x + 2) + 2)/(2(-5x + 2) - 1))^2 + 5((-2(-5x + 2) + 2)/(2(-5x + 2) - 1)) - 4))) + 2| + 1 and g(x) = 4(-1|(-1(5x + 1) + 3)... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-2|(3(1((-2x + 2)/(-3x))^2 - 3((-2x + 2)/(-3x)) - 1) - 4) + 2| + 2) + 1| + 2 and g(x) = (-1(-3(-1((3x - 2)/(1x + 1)) + 3)^2 - 3(-1((3x - 2)/(1x + 1)) + 3) + 3) - 3)/(-3(-3(-1((3x - 2)/(1x + 1)) + 3)^2 - 3(-1((3x - 2)/(1x + 1)) ... | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-1((-3(2sin(1π(3x^2 + 4x - 2) + 2) + 3) - 2)/(-2(2sin(1π(3x^2 + 4x - 2) + 2) + 3) + 3))^2 - 3((-3(2sin(1π(3x^2 + 4x - 2) + 2) + 3) - 2)/(-2(2sin(1π(3x^2 + 4x - 2) + 2) + 3) + 3)) + 1) - 3| + 1 and g(x) = (2(1|(-1|((-2x)/(-1x + 3)) - 2| - 2) + 3| + 2... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(-3|(-3((2(-2cos(1πx) + 1))/(2(-2cos(1πx) + 1) + 3))^2 + 2((2(-2cos(1πx) + 1))/(2(-2cos(1πx) + 1) + 3)) + 5) + 1| - 3) + 1| + 2 and g(x) = -1|((2(2|(3x^2 + 2x - 2) + 3| - 1) - 2)/(-1(2|(3x^2 + 2x - 2) + 3| - 1) + 2)) - 2| ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(2(-1sin(1π(1(3x - 4)^2 - 3(3x - 4) + 4) + 1))^2 + 4(-1sin(1π(1(3x - 4)^2 - 3(3x - 4) + 4) + 1)) - 1) + 2 and g(x) = -1|(-5(-2(-1x + 2)^2 - 2(-1x + 2) + 5)) - 3| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format... | 61 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(3|((-3(1cos(1π(5x^2 - 2x + 4) - 1) + 2) + 2)/(-2(1cos(1π(5x^2 - 2x + 4) - 1) + 2) + 1)) - 1| - 1)^2 + 1(3|((-3(1cos(1π(5x^2 - 2x + 4) - 1) + 2) + 2)/(-2(1cos(1π(5x^2 - 2x + 4) - 1) + 2) + 1)) - 1| - 1) + 1 and g(x) = (-1(3cos(2π(-3|(-1x - 5) - 2| + ... | 793 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-2(-3|((-3(-3x + 2) - 3)/(-2(-3x + 2) - 2)) - 1| - 2) - 4) + 1| - 1 and g(x) = -3(5(-2|(3|x + 3|) - 1|) + 1) + 3 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) = -5(3cos(3π((1(5(-2x - 5) + 5) - 3)/(-3(5(-2x - 5) + 5) + 2)) - 2)) - 5 and g(x) = 2(-3sin(3π(5(1x + 5) + 1) + 3) - 1)^2 - 5(-3sin(3π(5(1x + 5) + 1) + 3) - 1) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\bo... | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-2|((3(-4(-2x - 3)^2 - 5(-2x - 3) + 1) - 3)/(2(-4(-2x - 3)^2 - 5(-2x - 3) + 1) + 3)) + 2| + 3) - 2| + 2 and g(x) = (1(3|((2(1x - 5) + 2)/(-2(1x - 5) + 1)) + 1| - 2) - 3)/(-1(3|((2(1x - 5) + 2)/(-2(1x - 5) + 1)) + 1| - 2) - 3) intersect in the interv... | 7 | MATH |
[
{
"content": "How many solutions does the equation (1(5((2(1|(2cos(2πx)) - 3| - 2) + 1)/(1(1|(2cos(2πx)) - 3| - 2)))^2 - 2((2(1|(2cos(2πx)) - 3| - 2) + 1)/(1(1|(2cos(2πx)) - 3| - 2))) - 4) - 3)/(-3(5((2(1|(2cos(2πx)) - 3| - 2) + 1)/(1(1|(2cos(2πx)) - 3| - 2)))^2 - 2((2(1|(2cos(2πx)) - 3| - 2) + 1)/(1(1|(2cos(2π... | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(2cos(3π(-2(-2|(-4x - 1) + 1|)^2 - 4(-2|(-4x - 1) + 1|) + 5) + 3) - 3) - 3)/(-3(2cos(3π(-2(-2|(-4x - 1) + 1|)^2 - 4(-2|(-4x - 1) + 1|) + 5) + 3) - 3) - 3) and g(x) = 5(-2|(-3cos(3π(4x + 2) + 2) - 2) + 3|)^2 + 1(-2|(-3cos(3π(4x +... | 300 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5((2(2cos(3π(3(2x + 1)^2 - 4(2x + 1) + 1) + 1) - 1))/(-3(2cos(3π(3(2x + 1)^2 - 4(2x + 1) + 1) + 1) - 1))) + 2 and g(x) = -1(2(2cos(3π(-4x - 2) + 2) - 3) + 1) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\b... | 240 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((-3(2cos(1π((2(-4x - 5) - 2)/(1(-4x - 5) + 3)) + 1)) + 1)/(-2(2cos(1π((2(-4x - 5) - 2)/(1(-4x - 5) + 3)) + 1)))) + 4 and g(x) = -1((-1(-3(3x + 2) + 2) + 3)/(-2(-3(3x + 2) + 2) - 3))^2 + 4((-1(-3(3x + 2) + 2) + 3)/(-2(-3(3x... | 15 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2((3(4(5((-3x + 2)/(2x - 1))^2 - 2((-3x + 2)/(2x - 1)) - 2) + 5) - 1)/(-2(4(5((-3x + 2)/(2x - 1))^2 - 2((-3x + 2)/(2x - 1)) - 2) + 5) + 3)) and g(x) = -4(1|(-3(1cos(2πx + 1) - 1) - 1) + 3| - 1)^2 + 1(1|(-3(1cos(2πx + 1) - 1) ... | 41 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((-2(-3|((1(1sin(3πx + 3) - 1) + 1)/(-1(1sin(3πx + 3) - 1) - 1))| + 2))/(2(-3|((1(1sin(3πx + 3) - 1) + 1)/(-1(1sin(3πx + 3) - 1) - 1))| + 2) - 1))^2 - 3((-2(-3|((1(1sin(3πx + 3) - 1) + 1)/(-1(1sin(3πx + 3) - 1) - 1))| + 2))/(2(-3... | 81 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(1((-2(1cos(3π(-1x - 4) + 2)) + 3)/(-1(1cos(3π(-1x - 4) + 2)) - 2))^2 + 1((-2(1cos(3π(-1x - 4) + 2)) + 3)/(-1(1cos(3π(-1x - 4) + 2)) - 2)) - 1) - 2| - 1 and g(x) = -5(1cos(2π((2(5x^2 - 4x + 2) + 3)/(-3(5x^2 - 4x + 2))) - 1... | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|((1((3((-2(-2x^2 + 1x - 1) + 1)/(-3(-2x^2 + 1x - 1) - 2)) - 3)/(1((-2(-2x^2 + 1x - 1) + 1)/(-3(-2x^2 + 1x - 1) - 2)) - 3)) + 2)/(3((3((-2(-2x^2 + 1x - 1) + 1)/(-3(-2x^2 + 1x - 1) - 2)) - 3)/(1((-2(-2x^2 + 1x - 1) + 1)/(-3(-2x^2 +... | 48 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(3|(-1|((-2(-3x + 3) - 2)/(3(-3x + 3) - 2)) - 3| - 2) + 1| + 1) - 3)/(1(3|(-1|((-2(-3x + 3) - 2)/(3(-3x + 3) - 2)) - 3| - 2) + 1| + 1) + 1) and g(x) = 2(-1cos(3π((-2(2x^2 - 3x - 3) + 3)/(1(2x^2 - 3x - 3) + 3)) + 3) - 1)^2 - 2(-1cos(3π((-2(2x^2 - 3x -... | 155 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(3cos(2π(1(-3|(1x^2 + 5x + 1) - 1| - 1) - 5) - 3) - 1) - 1 and g(x) = -3sin(2π(2|(3(-5x - 5) - 2) - 1| - 1)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1103 | MATH |
[
{
"content": "How many solutions does the equation 5((-2(3cos(1π(-1|(-4x^2 - 3x - 5) - 2|) - 2)) + 2)/(-2(3cos(1π(-1|(-4x^2 - 3x - 5) - 2|) - 2)) - 3))^2 + 2((-2(3cos(1π(-1|(-4x^2 - 3x - 5) - 2|) - 2)) + 2)/(-2(3cos(1π(-1|(-4x^2 - 3x - 5) - 2|) - 2)) - 3)) = 3((2(1|(5x^2 + 5x) + 2| + 2) + 3)/(2(1|(5x^2 + 5x) + ... | 704 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(1|(-2((1((-1x - 2)/(2x)) - 2)/(2((-1x - 2)/(2x)) + 3))^2 - 1((1((-1x - 2)/(2x)) - 2)/(2((-1x - 2)/(2x)) + 3)) - 2) - 3| - 2) + 2| + 1 and g(x) = -4(1|(-5(-4x - 4) - 3)|)^2 + 5(1|(-5(-4x - 4) - 3)|) - 3 in the range -10 ≤ x ≤ 10... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2((1((-1(-2(-1cos(3πx + 3) - 3)^2 + 5(-1cos(3πx + 3) - 3) + 4))/(-1(-2(-1cos(3πx + 3) - 3)^2 + 5(-1cos(3πx + 3) - 3) + 4) - 3)) - 2)/(-3((-1(-2(-1cos(3πx + 3) - 3)^2 + 5(-1cos(3πx + 3) - 3) + 4))/(-1(-2(-1cos(3πx + 3) - 3)^2 + 5(-1cos(3πx + 3) - 3) + 4) - 3)) -... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-3cos(2π(-1((-3(3|x + 3| + 3) - 2)/(-3(3|x + 3| + 3) + 4)) - 3) - 1) - 3) + 3 and g(x) = 3((-2(-4(3|x + 3| - 1)))/(3(-4(3|x + 3| - 1)) - 2))^2 - 1((-2(-4(3|x + 3| - 1)))/(3(-4(3|x + 3| - 1)) - 2)) - 5 in the range -10 ≤ x ≤ 10?\n... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-1(1(-3(-2sin(3πx - 1) - 1)^2 + 1(-2sin(3πx - 1) - 1) + 4) + 4)^2 - 1(1(-3(-2sin(3πx - 1) - 1)^2 + 1(-2sin(3πx - 1) - 1) + 4) + 4) - 2) - 4 and g(x) = -3(-3(4(-5x^2 - 4x + 1)^2 - 2(-5x^2 - 4x + 1) - 1) + 4)^2 + 5(-3(4(-5x^2 - 4x ... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(3π(-2|(4((-2((2x - 3)/(2x)) - 2)/(-1((2x - 3)/(2x)) - 3))^2 - 5((-2((2x - 3)/(2x)) - 2)/(-1((2x - 3)/(2x)) - 3)) + 5) - 2| - 3) - 3) and g(x) = 5(1(-3cos(2π((-2x + 2)/(2x + 3)) - 1))^2 + 5(-3cos(2π((-2x + 2)/(2x + 3)) - 1)) + 3)^2 - 1(1(-3cos(2π((... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-3|(-2sin(1π(-5((2x - 1)/(-2x - 1))^2 + 1((2x - 1)/(-2x - 1)) + 2) + 3) + 3) - 2|)^2 - 1(-3|(-2sin(1π(-5((2x - 1)/(-2x - 1))^2 + 1((2x - 1)/(-2x - 1)) + 2) + 3) + 3) - 2|) - 2 and g(x) = 5((-2((-2(-1x - 1))/(3(-1x - 1) - 2)... | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((1(-5(-2|(-1cos(2πx - 1) + 1) - 2| + 3) - 2))/(3(-5(-2|(-1cos(2πx - 1) + 1) - 2| + 3) - 2) + 1)) - 4 and g(x) = 3(1e^(-1((-3((2x + 2)/(1x + 3)) + 1)/(2((2x + 2)/(1x + 3)) + 2))) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in L... | 27 | MATH |
[
{
"content": "How many solutions does the equation 3sin(1π(2((3(-1|(-1x^2 - 1x - 1) - 1| - 2) - 2)/(-3(-1|(-1x^2 - 1x - 1) - 1| - 2)))^2 - 2((3(-1|(-1x^2 - 1x - 1) - 1| - 2) - 2)/(-3(-1|(-1x^2 - 1x - 1) - 1| - 2))) - 3) + 2) + 2 = 1(-3|(5(-3e^(1x) + 2) + 3) - 2| + 3)^2 - 3(-3|(5(-3e^(1x) + 2) + 3) - 2| + 3) + 5... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3(-2(1(3x^2 - 3x + 1)^2 + 4(3x^2 - 3x + 1) - 4) + 1) - 1)^2 - 2(-3(-2(1(3x^2 - 3x + 1)^2 + 4(3x^2 - 3x + 1) - 4) + 1) - 1) and g(x) = (-3((2(4(3|x + 2| + 3) + 1) - 3)/(1(4(3|x + 2| + 3) + 1) + 2)) + 2)/(-2((2(4(3|x + 2| + 3) + 1) - 3)/(1(4(3|x + 2| ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-1|(-2(-3|((3x - 1)/(3x + 3)) + 2|)^2 + 5(-3|((3x - 1)/(3x + 3)) + 2|)) - 1| + 2) - 1 and g(x) = 3e^(2(3(1|(2x^2 + 1x - 1) - 3| - 1) + 4)) + 1 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) = 1(-2|(-2sin(3π((3(4x^2 - 4x))/(-3(4x^2 - 4x) + 1)) + 1) - 3) - 2|)^2 + 5(-2|(-2sin(3π((3(4x^2 - 4x))/(-3(4x^2 - 4x) + 1)) + 1) - 3) - 2|) + 2 and g(x) = 4(1(1|(1x - 5) - 2| - 2))^2 - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answ... | 35 | MATH |
[
{
"content": "How many solutions does the equation -5(-1((-1(2(1x^2 + 4x - 4)^2 + 5(1x^2 + 4x - 4) + 2) + 1)/(-2(2(1x^2 + 4x - 4)^2 + 5(1x^2 + 4x - 4) + 2) + 1)) - 1)^2 - 4(-1((-1(2(1x^2 + 4x - 4)^2 + 5(1x^2 + 4x - 4) + 2) + 1)/(-2(2(1x^2 + 4x - 4)^2 + 5(1x^2 + 4x - 4) + 2) + 1)) - 1) - 2 = -2|(3((-2((3x - 2)/(... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1(4(-3(2|x - 1| - 2)) - 4) - 5) - 1 and g(x) = 4(1sin(3π(3|(-2x^2 - 1x - 3)|) - 3))^2 - 5(1sin(3π(3|(-2x^2 - 1x - 3)|) - 3)) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(3π((1(2|(-2(5x^2 + 2x - 3)^2 - 1(5x^2 + 2x - 3) + 3) + 2| + 1) - 2)/(-1(2|(-2(5x^2 + 2x - 3)^2 - 1(5x^2 + 2x - 3) + 3) + 2| + 1) + 2)) - 1) + 2 and g(x) = 1|(-4(-3e^(1((3x + 3)/(-1x + 3))) + 2)^2 - 3(-3e^(1((3x + 3)/(-1x + 3))... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4((1((3(4(1e^(-1x) - 1)) + 2)/(1(4(1e^(-1x) - 1)) + 1)) + 2)/(-3((3(4(1e^(-1x) - 1)) + 2)/(1(4(1e^(-1x) - 1)) + 1)) + 2)) - 1 and g(x) = 2|(-2|(2(-5x + 5)^2 - 4(-5x + 5)) + 2| + 1)| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex f... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3e^(1((1((3(3(-5x^2 + 5x + 2)^2 - 1(-5x^2 + 5x + 2) + 1) + 1)/(-1(3(-5x^2 + 5x + 2)^2 - 1(-5x^2 + 5x + 2) + 1) - 3)) - 1)/(3((3(3(-5x^2 + 5x + 2)^2 - 1(-5x^2 + 5x + 2) + 1) + 1)/(-1(3(-5x^2 + 5x + 2)^2 - 1(-5x^2 + 5x + 2) + 1) - 3)... | 244 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1cos(1π(2((2(3x^2 - 5x + 4) + 3)/(-2(3x^2 - 5x + 4) + 3))^2 - 5((2(3x^2 - 5x + 4) + 3)/(-2(3x^2 - 5x + 4) + 3)) - 1) - 2) + 3)^2 + 3(-1cos(1π(2((2(3x^2 - 5x + 4) + 3)/(-2(3x^2 - 5x + 4) + 3))^2 - 5((2(3x^2 - 5x + 4) + 3)/(... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(4(1sin(1π((1(-4x) - 2)/(-1(-4x) + 2)) - 2) + 1)^2 - 5(1sin(1π((1(-4x) - 2)/(-1(-4x) + 2)) - 2) + 1) + 2) + 2)/(-3(4(1sin(1π((1(-4x) - 2)/(-1(-4x) + 2)) - 2) + 1)^2 - 5(1sin(1π((1(-4x) - 2)/(-1(-4x) + 2)) - 2) + 1) + 2)) and g(x)... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-1((2((-1(3sin(1πx + 1) + 2))/(2(3sin(1πx + 1) + 2) + 3)) + 2)/(2((-1(3sin(1πx + 1) + 2))/(2(3sin(1πx + 1) + 2) + 3)) - 3)) - 5) + 1| - 2 and g(x) = -2cos(2π(1|(1(-4x)^2 - 5(-4x) + 2) - 2| - 2) - 1) + 2 intersect in the interval [-10, 10].\n\nPresen... | 1296 | MATH |
[
{
"content": "How many solutions does the equation -4(-4((2(-1sin(1π(-5x^2 + 5x + 2) + 2)) - 3)/(-1(-1sin(1π(-5x^2 + 5x + 2) + 2)) - 3)) + 2) + 1 = (2(-1sin(2π(1|((-1x - 2)/(-1x - 3)) - 1| - 1) + 1) - 3) - 3)/(-3(-1sin(2π(1|((-1x - 2)/(-1x - 3)) - 1| - 1) + 1) - 3) - 3) have for x in the interval [-10, 10]?\n\n... | 815 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(3|(2|((3(-1|x + 2| + 3) - 1)/(-1(-1|x + 2| + 3) + 3)) - 3| - 2) - 2| - 2)| - 1 and g(x) = (-2((-3((1(2|x + 1| + 1) - 1)/(-2(2|x + 1| + 1) - 3)) + 2)/(2((1(2|x + 1| + 1) - 1)/(-2(2|x + 1| + 1) - 3)))) + 3)/(-2((-3((1(2|x + 1| + 1) - 1)/(-2(2|x + 1| +... | 16 | MATH |
[
{
"content": "How many solutions does the equation -1|(-3|(-1(1(2cos(3πx) - 1) - 3)^2 + 1(1(2cos(3πx) - 1) - 3) - 5) + 3| - 2) - 2| - 3 = (-2(2((-3(-3cos(1πx + 3) + 1) - 3)/(2(-3cos(1πx + 3) + 1)))^2 + 3((-3(-3cos(1πx + 3) + 1) - 3)/(2(-3cos(1πx + 3) + 1))) - 3) + 3)/(-3(2((-3(-3cos(1πx + 3) + 1) - 3)/(2(-3cos(... | 8 | MATH |
[
{
"content": "How many solutions does the equation (-3(-1(-1|(-2|((3x + 3)/(-1x + 2)) - 1| + 1) - 2| + 3)^2 - 5(-1|(-2|((3x + 3)/(-1x + 2)) - 1| + 1) - 2| + 3) - 5))/(1(-1(-1|(-2|((3x + 3)/(-1x + 2)) - 1| + 1) - 2| + 3)^2 - 5(-1|(-2|((3x + 3)/(-1x + 2)) - 1| + 1) - 2| + 3) - 5) + 2) = -2|(3(-4(-2sin(1πx - 2)) +... | 45 | MATH |
[
{
"content": "How many solutions does the equation -2sin(2π(-4(-1(5((3x)/(3x - 2)) - 4) + 1)^2 - 5(-1(5((3x)/(3x - 2)) - 4) + 1)) - 3) + 3 = 3sin(3π((-3((1(-4x + 1) - 1)/(2(-4x + 1) - 3)))/(-1((1(-4x + 1) - 1)/(2(-4x + 1) - 3)) + 2)) - 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex ... | 186 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((3(1|(1((3x + 1)/(1x)) + 4) - 1| - 2) + 3)/(1(1|(1((3x + 1)/(1x)) + 4) - 1| - 2) - 2)) - 1)/(-1((3(1|(1((3x + 1)/(1x)) + 4) - 1| - 2) + 3)/(1(1|(1((3x + 1)/(1x)) + 4) - 1| - 2) - 2))) and g(x) = -3cos(1π(3|((3(-1|x - 3| - 1) + 1... | 8 | MATH |
[
{
"content": "How many solutions does the equation 3|((3(-2cos(1π((-2(-1x^2 - 4x + 5) - 3)/(-3(-1x^2 - 4x + 5) + 2))) - 1) + 3)/(1(-2cos(1π((-2(-1x^2 - 4x + 5) - 3)/(-3(-1x^2 - 4x + 5) + 2))) - 1))) + 3| - 3 = 2cos(3π((-1((-2(-4x) + 3)/(2(-4x) - 1)) + 3)/(2((-2(-4x) + 3)/(2(-4x) - 1)) + 2)) - 1) have for x in t... | 44 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(2|(-1sin(2π(3(2x^2)^2 - 3(2x^2) - 4)) - 3) - 1| - 2))/(-3(2|(-1sin(2π(3(2x^2)^2 - 3(2x^2) - 4)) - 3) - 1| - 2) - 3) and g(x) = -3((3(-3(-3x^2 - 2x + 1)^2 - 4(-3x^2 - 2x + 1) + 5) + 1)/(2(-3(-3x^2 - 2x + 1)^2 - 4(-3x^2 - 2x + 1)... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-1(-1|((-1(4x - 4) - 2)/(-2(4x - 4) + 3)) + 3| + 2)^2 + 5(-1|((-1(4x - 4) - 2)/(-2(4x - 4) + 3)) + 3| + 2) + 3) - 2 and g(x) = 2(1sin(2π(-2(3|x + 1| - 1)^2 - 5(3|x + 1| - 1) + 2) - 2) + 2)^2 - 2(1sin(2π(-2(3|x + 1| - 1)^2 - 5(3|x... | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2e^(1(-1|((1(-3|(-2|x + 2| - 3) + 1| + 2))/(-3(-3|(-2|x + 2| - 3) + 1| + 2) + 1)) + 1| - 2)) + 2 and g(x) = (2((-1(2(3cos(3πx + 2) + 2)^2 - 5(3cos(3πx + 2) + 2) - 4) + 2)/(-2(2(3cos(3πx + 2) + 2)^2 - 5(3cos(3πx + 2) + 2) - 4... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(2|((-2(1((-1x + 1)/(2x + 2))^2 - 2((-1x + 1)/(2x + 2)) - 2) - 3)/(-1(1((-1x + 1)/(2x + 2))^2 - 2((-1x + 1)/(2x + 2)) - 2) - 3)) - 3| - 3)^2 - 3 and g(x) = (-3(5(-3(-1x^2 - 5x - 5) + 2)^2 - 3(-3(-1x^2 - 5x - 5) + 2) - 3) - 1... | 7 | MATH |
[
{
"content": "How many solutions does the equation 1((-2(-1(-5(4x^2 - 2x + 4)^2 - 5(4x^2 - 2x + 4) - 3)^2 - 3(-5(4x^2 - 2x + 4)^2 - 5(4x^2 - 2x + 4) - 3) - 4) + 2)/(3(-1(-5(4x^2 - 2x + 4)^2 - 5(4x^2 - 2x + 4) - 3)^2 - 3(-5(4x^2 - 2x + 4)^2 - 5(4x^2 - 2x + 4) - 3) - 4))) = -2(-4(-5(5x^2 + 5x + 2)^2 + 3(5x^2 + 5x... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-2(2sin(3π(-5((2x - 1)/(1x + 1)) + 5) - 1) - 3)^2 - 2(2sin(3π(-5((2x - 1)/(1x + 1)) + 5) - 1) - 3) - 4)^2 + 5(-2(2sin(3π(-5((2x - 1)/(1x + 1)) + 5) - 1) - 3)^2 - 2(2sin(3π(-5((2x - 1)/(1x + 1)) + 5) - 1) - 3) - 4) + 3 and g(x) =... | 546 | MATH |
[
{
"content": "How many solutions does the equation -2|(1|(-1(1(1x + 5)^2 - 1(1x + 5) + 1) - 4)| + 2) + 2| - 1 = -1|(-1(-2|(3sin(1πx - 2) - 3) + 2| - 3)^2 - 4(-2|(3sin(1πx - 2) - 3) + 2| - 3) - 5) + 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|((2(3cos(3π(-3(1x^2 - 2x - 2)^2 + 3(1x^2 - 2x - 2) - 2) + 1)) + 3)/(-2(3cos(3π(-3(1x^2 - 2x - 2)^2 + 3(1x^2 - 2x - 2) - 2) + 1)) - 2))| and g(x) = -4((-3(-2e^(1(5x^2 + 3x + 1))) - 2)/(1(-2e^(1(5x^2 + 3x + 1))) + 3)) - 5 in the ra... | 153 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1|((-1(2(-3x) - 2) + 1)/(1(2(-3x) - 2) + 1)) + 2| - 1)^2 + 4(1|((-1(2(-3x) - 2) + 1)/(1(2(-3x) - 2) + 1)) + 2| - 1) - 1 and g(x) = -1|(2(3|(-3x + 4) - 1| - 3)^2 - 1(3|(-3x + 4) - 1| - 3) + 5) - 3| + 1 where -10 ≤ x ≤ 10.\n... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(5((-3((-3((1x + 3)/(1x + 1)) - 1)/(-1((1x + 3)/(1x + 1)) - 1)) + 3)/(1((-3((1x + 3)/(1x + 1)) - 1)/(-1((1x + 3)/(1x + 1)) - 1)) - 2))^2 + 5((-3((-3((1x + 3)/(1x + 1)) - 1)/(-1((1x + 3)/(1x + 1)) - 1)) + 3)/(1((-3((1x + 3)/(1x + 1)) - 1)/(-1((1x + 3)... | 1299 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(3π((1((-1(-2(2|x + 3| - 3) - 1) - 1)/(-1(-2(2|x + 3| - 3) - 1) - 2)) + 1)/(-3((-1(-2(2|x + 3| - 3) - 1) - 1)/(-1(-2(2|x + 3| - 3) - 1) - 2)))) - 1) and g(x) = 1((-1(-3(-3|x| - 2) - 4) - 1)/(1(-3(-3|x| - 2) - 4) + 1)) + 2 intersect in the interval ... | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(1|(5(-1(-1cos(1πx + 3) + 3) + 5)^2 + 3) - 3| - 3) - 5 and g(x) = 3|(-4((1(-5x^2 - 4x - 4) - 2)/(3(-5x^2 - 4x - 4) + 1))^2 - 5((1(-5x^2 - 4x - 4) - 2)/(3(-5x^2 - 4x - 4) + 1)) - 2) - 1| - 2 where -10 ≤ x ≤ 10.\n\nPresent the... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(1π((-2(-2(4(3x^2 + 4x + 1) - 2)) + 3)/(1(-2(4(3x^2 + 4x + 1) - 2)) - 3)) + 2) - 1 and g(x) = (2((-3(2(5x - 1)^2 + 5(5x - 1) - 2) + 2)/(-1(2(5x - 1)^2 + 5(5x - 1) - 2) + 3)) - 2)/(1((-3(2(5x - 1)^2 + 5(5x - 1) - 2) + 2)/(-1(2(5x - 1)^2 + 5(5x - 1) ... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(3((-3(3sin(2π(1x + 1) - 3) - 1) - 3)/(-2(3sin(2π(1x + 1) - 3) - 1) + 3))^2 + 3((-3(3sin(2π(1x + 1) - 3) - 1) - 3)/(-2(3sin(2π(1x + 1) - 3) - 1) + 3)) - 5)| - 2 and g(x) = -1|(-1(1|(-2cos(2πx - 2) + 3) - 3| - 3) + 5) - 2| ... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-1e^(2((3((-1(-1x^2 - 5x - 2) - 2)/(1(-1x^2 - 5x - 2) + 1)) + 1)/(-3((-1(-1x^2 - 5x - 2) - 2)/(1(-1x^2 - 5x - 2) + 1)) + 1)))) + 5 and g(x) = -2(-1(2((-2x)/(-1x + 3))^2 + 2((-2x)/(-1x + 3)) - 5) - 4)^2 - 5(-1(2((-2x)/(-1x + 3))^2... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(3(-2sin(1π(1|(-5x + 4) + 3| + 2) + 3) - 1)^2 + 2(-2sin(1π(1|(-5x + 4) + 3| + 2) + 3) - 1) - 5) + 3 and g(x) = 2(-3(1(-1|x - 3| + 1)^2 - 2(-1|x - 3| + 1) - 3)^2 - 4)^2 - 1(-3(1(-1|x - 3| + 1)^2 - 2(-1|x - 3| + 1) - 3)^2 - 4) - 3 intersect in the inter... | 4 | MATH |
[
{
"content": "How many solutions does the equation -4((1((-3((-1((1x - 2)/(-2x - 3)))/(-2((1x - 2)/(-2x - 3)) - 3)) - 2)/(1((-1((1x - 2)/(-2x - 3)))/(-2((1x - 2)/(-2x - 3)) - 3)) - 2)) - 2)/(-2((-3((-1((1x - 2)/(-2x - 3)))/(-2((1x - 2)/(-2x - 3)) - 3)) - 2)/(1((-1((1x - 2)/(-2x - 3)))/(-2((1x - 2)/(-2x - 3)) - ... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(4(-3|((2(5x^2 - 1x - 1) + 3)/(-3(5x^2 - 1x - 1) + 4)) + 3| + 2) + 2) - 2)/(1(4(-3|((2(5x^2 - 1x - 1) + 3)/(-3(5x^2 - 1x - 1) + 4)) + 3| + 2) + 2) + 1) and g(x) = 3((2(-2e^(-1((-3x + 3)/(2x))) - 1) + 1)/(3(-2e^(-1((-3x + 3)/(2x))) - 1) - 1)) + 3 inte... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(1(3(1sin(1π((-3x - 2)/(3x - 1)) + 2) + 2))^2 + 3(3(1sin(1π((-3x - 2)/(3x - 1)) + 2) + 2)) - 5) - 3 and g(x) = -3|(-4(-3(-2x + 1) - 3)^2 + 1(-3(-2x + 1) - 3) + 4) - 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex form... | 33 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-3cos(2π((1(-1(4x - 3)^2 + 3(4x - 3) - 1) - 3)/(-2(-1(4x - 3)^2 + 3(4x - 3) - 1) - 3)) - 3) + 2) + 3)/(-3(-3cos(2π((1(-1(4x - 3)^2 + 3(4x - 3) - 1) - 3)/(-2(-1(4x - 3)^2 + 3(4x - 3) - 1) - 3)) - 3) + 2) - 1) and g(x) = (-1(2sin(3π((2(-4x^2 + 1x + 2... | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(1((3(-2|(1x^2 - 1x - 3) - 3| + 2) + 1)/(-1(-2|(1x^2 - 1x - 3) - 3| + 2) - 2)) - 5) - 1 and g(x) = -3|(-5(2(-1x^2 + 1x + 2) - 3))| - 3 in the range -10 ≤ x ≤ 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) = 5(2((2((3(-2|x - 1| + 1) - 1)/(-3(-2|x - 1| + 1))) - 1)/(-2((3(-2|x - 1| + 1) - 1)/(-3(-2|x - 1| + 1))) - 3))^2 + 5((2((3(-2|x - 1| + 1) - 1)/(-3(-2|x - 1| + 1))) - 1)/(-2((3(-2|x - 1| + 1) - 1)/(-3(-2|x - 1| + 1))) - 3)) + 1... | 4 | MATH |
[
{
"content": "How many solutions does the equation -5((3((-3(-1(-2|x - 2|)^2 + 2(-2|x - 2|) - 4))/(1(-1(-2|x - 2|)^2 + 2(-2|x - 2|) - 4) - 3)) - 3)/(2((-3(-1(-2|x - 2|)^2 + 2(-2|x - 2|) - 4))/(1(-1(-2|x - 2|)^2 + 2(-2|x - 2|) - 4) - 3)) + 3)) - 2 = (3(-2|(-3cos(2π(-1|x - 2| - 3) + 1) - 2) + 2|))/(-2(-2|(-3cos(2... | 37 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-3|(-3((1((-1x + 1)/(2x + 2)) + 2)/(-2((-1x + 1)/(2x + 2)) - 3))^2 - 1((1((-1x + 1)/(2x + 2)) + 2)/(-2((-1x + 1)/(2x + 2)) - 3)) - 1) + 2| + 1) + 2 and g(x) = (-1(-4(2((-3x - 3)/(-2x - 2))^2 + 1((-3x - 3)/(-2x - 2)) - 2)^2 + 2(2((-3x - 3)/(-2x - 2))... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(5(-3sin(2π(1(-1x^2 + 4x)^2 - 2(-1x^2 + 4x) - 3) + 3) - 1) - 5) - 2| - 2 and g(x) = -1sin(2π(-2|(3(-2x^2 - 2) + 4) + 2| - 2) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 150 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(2|(-3|(-3sin(3π(4x - 4) - 1) - 1) - 2|) - 3| + 3) + 1 and g(x) = 1(2(-5(3cos(2πx - 3) - 1)^2 - 2(3cos(2πx - 3) - 1) - 2) - 1)^2 + 4(2(-5(3cos(2πx - 3) - 1)^2 - 2(3cos(2πx - 3) - 1) - 2) - 1) + 3 intersect in the interval [-10, 10].\n\nPresent the ans... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(2cos(2π(-1(-2(5x^2 - 1x + 5)^2 + 5(5x^2 - 1x + 5) - 4)^2 - 4(-2(5x^2 - 1x + 5)^2 + 5(5x^2 - 1x + 5) - 4) + 1) + 3) + 1) - 3 and g(x) = -3cos(2π(5((-2(-5x - 4) + 1)/(-2(-5x - 4))) - 1) - 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresen... | 135 | MATH |
[
{
"content": "How many solutions does the equation -5((3(1(-4(1x^2 + 4x + 5) + 4)^2 - 4(-4(1x^2 + 4x + 5) + 4) - 3))/(2(1(-4(1x^2 + 4x + 5) + 4)^2 - 4(-4(1x^2 + 4x + 5) + 4) - 3) - 1))^2 + 3((3(1(-4(1x^2 + 4x + 5) + 4)^2 - 4(-4(1x^2 + 4x + 5) + 4) - 3))/(2(1(-4(1x^2 + 4x + 5) + 4)^2 - 4(-4(1x^2 + 4x + 5) + 4) -... | 679 | MATH |
[
{
"content": "How many solutions does the equation 2(-2|(-2(1(-1x + 1) - 3)^2 - 3(1(-1x + 1) - 3) - 2)|) + 4 = -3(2(-1|(-3x^2 + 4) - 1| + 3)^2 + 5(-1|(-3x^2 + 4) - 1| + 3) - 3) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|((2(2e^(1(4(1x + 4)^2 - 3(1x + 4) - 5)) - 3) - 2)/(-2(2e^(1(4(1x + 4)^2 - 3(1x + 4) - 5)) - 3) - 1)) - 1| + 2 and g(x) = -1|(4(2(3sin(2πx + 1) + 1)^2 + 2(3sin(2πx + 1) + 1) + 2) - 3) - 3| in the range -10 ≤ x ≤ 10?\n\nPresent th... | 43 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(1|((-3(-1(-5x + 3)^2 + 4(-5x + 3) - 2) - 1)/(-2(-1(-5x + 3)^2 + 4(-5x + 3) - 2) + 1)) - 2| - 2) - 1| - 3 and g(x) = 1(2sin(3π(-1(-5x^2 - 4x - 1) - 5) + 2))^2 - 3(2sin(3π(-1(-5x^2 - 4x - 1) - 5) + 2)) + 4 in the range -10 ≤ x ≤ 1... | 1616 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-2(-2(2|(2x^2 - 1x - 4) + 2| - 1)^2 - 4(2|(2x^2 - 1x - 4) + 2| - 1) + 5)^2 + 2(-2(2|(2x^2 - 1x - 4) + 2| - 1)^2 - 4(2|(2x^2 - 1x - 4) + 2| - 1) + 5) + 5) and g(x) = -3|(-3|(-2|(2x^2 + 2x + 2) - 2| - 2)| - 1) + 2| in the range -10... | 8 | MATH |
[
{
"content": "How many solutions does the equation 1e^(-1((2(2(3|(2|x - 1| + 3) + 1| - 3) - 2) - 2)/(3(2(3|(2|x - 1| + 3) + 1| - 3) - 2) + 1))) - 1 = -1((-1(-3|(-3|x - 1| + 3) - 1| + 3) + 1)/(-3(-3|(-3|x - 1| + 3) - 1| + 3) - 1)) - 4 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) = -2(2cos(1π(-4(1(-1|x - 3| - 3) - 5)^2 - 2) - 3)) + 5 and g(x) = -1(-3cos(2π(1|(1x^2 - 5x)| + 1) - 2) - 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 807 | MATH |
[
{
"content": "How many solutions does the equation 5(-5((3(-2(4x - 3) + 3) - 1)/(2(-2(4x - 3) + 3) - 1))^2 - 3((3(-2(4x - 3) + 3) - 1)/(2(-2(4x - 3) + 3) - 1)) - 2) - 3 = -1e^(-1((3(-2(-3|x| + 1) - 2) + 3)/(-2(-2(-3|x| + 1) - 2) + 1))) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex for... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(3π(-4(-1((-2(-1|x + 1| - 3) - 2)/(-1(-1|x + 1| - 3) + 3)) - 1) + 3)) - 1 and g(x) = -5(3sin(2π(3(3x^2 + 1x - 5) - 2) + 3)) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 1657 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(1e^(2((3((3(-1|x + 2| + 2) + 3)/(3(-1|x + 2| + 2) - 2)) - 2)/(-3((3(-1|x + 2| + 2) + 3)/(3(-1|x + 2| + 2) - 2)) - 1))) + 3) - 1)/(1(1e^(2((3((3(-1|x + 2| + 2) + 3)/(3(-1|x + 2| + 2) - 2)) - 2)/(-3((3(-1|x + 2| + 2) + 3)/(... | 600 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(2(-3(-2|(1x - 3) + 2|)^2 - 2(-2|(1x - 3) + 2|) + 5)^2 - 4(-3(-2|(1x - 3) + 2|)^2 - 2(-2|(1x - 3) + 2|) + 5) + 2) - 1| + 1 and g(x) = -1(3sin(2π((-3((2x + 3)/(1x - 3)) + 2)/(3((2x + 3)/(1x - 3)) + 3)) - 1) + 3)^2 + 3(3sin(2... | 19 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(4(1((2(1e^(1x) - 3) + 3)/(2(1e^(1x) - 3))) - 2) + 3) + 3)/(3(4(1((2(1e^(1x) - 3) + 3)/(2(1e^(1x) - 3))) - 2) + 3) + 1) and g(x) = (3(-1|(-2cos(3π((-1x + 1)/(-3x - 1)) + 1) - 1) - 1| + 1) + 2)/(2(-1|(-2cos(3π((-1x + 1)/(-3x - 1)) + 1) - 1) - 1| + 1)... | 21 | MATH |
[
{
"content": "How many solutions does the equation -4((-2(-1|(-2((3x + 1)/(3x + 2)) - 2) + 1| + 2) + 3)/(2(-1|(-2((3x + 1)/(3x + 2)) - 2) + 1| + 2) - 1))^2 - 3((-2(-1|(-2((3x + 1)/(3x + 2)) - 2) + 1| + 2) + 3)/(2(-1|(-2((3x + 1)/(3x + 2)) - 2) + 1| + 2) - 1)) + 5 = (-3(-1|(1(2sin(1πx + 2) + 3)^2 + 5(2sin(1πx + ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(2(-1(-4(1|x - 1| - 2) - 4)^2 - 4(-4(1|x - 1| - 2) - 4) - 1) - 5) - 2| - 3 and g(x) = 1|(2sin(2π(2(1x + 3)^2 - 4(1x + 3)) + 3) + 3) - 1| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 16 | MATH |
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