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(1((-1(3|x + 1| - 1) + 1)/(-1(3|x + 1| - 1) + 3))^2 - 3((-1(3|x + 1| - 1) + 1)/(-1(3|x + 1| - 1) + 3)) - 4)^2 + 3(1((-1(3|x + 1| - 1) + 1)/(-1(3|x + 1| - 1) + 3))^2 - 3((-1(3|x + 1| - 1) + 1)/(-1(3|x + 1| - 1) + 3)) - 4)... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-2|(-3((-1((3x + 3)/(2x)) + 2)/(-1((3x + 3)/(2x)) + 3))^2 - 5((-1((3x + 3)/(2x)) + 2)/(-1((3x + 3)/(2x)) + 3)) + 3) + 1| + 1) + 3| - 2 and g(x) = -1((-1(2|((1x)/(-1x - 2)) - 3|) - 1)/(3(2|((1x)/(-1x - 2)) - 3|) - 2))^2 + 2((-1(2|((1x)/(-1x - 2)) - 3... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-3((2(1((-3x)/(-1x + 2))^2 + 1((-3x)/(-1x + 2)) + 4) - 3)/(1(1((-3x)/(-1x + 2))^2 + 1((-3x)/(-1x + 2)) + 4) + 3))^2 - 3((2(1((-3x)/(-1x + 2))^2 + 1((-3x)/(-1x + 2)) + 4) - 3)/(1(1((-3x)/(-1x + 2))^2 + 1((-3x)/(-1x + 2)) + 4) + ... | 34 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-3sin(2π(3((1(5x^2 - 1x - 3) + 1)/(1(5x^2 - 1x - 3)))^2 - 4((1(5x^2 - 1x - 3) + 1)/(1(5x^2 - 1x - 3))) - 1) - 3) + 3)^2 - 2(-3sin(2π(3((1(5x^2 - 1x - 3) + 1)/(1(5x^2 - 1x - 3)))^2 - 4((1(5x^2 - 1x - 3) + 1)/(1(5x^2 - 1x - ... | 58 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-1|(3(4(-1x + 1)^2 + 3(-1x + 1) + 3)^2 + 1(4(-1x + 1)^2 + 3(-1x + 1) + 3) + 1) - 1| - 1)^2 - 4(-1|(3(4(-1x + 1)^2 + 3(-1x + 1) + 3)^2 + 1(4(-1x + 1)^2 + 3(-1x + 1) + 3) + 1) - 1| - 1) + 3 and g(x) = 2|((-3(2cos(3π(-2x + 5) - 1) +... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-3cos(2π((1(-4((1x + 3)/(-2x - 2))^2 - 5((1x + 3)/(-2x - 2)) + 3))/(-2(-4((1x + 3)/(-2x - 2))^2 - 5((1x + 3)/(-2x - 2)) + 3) - 2)) + 2) + 3)^2 + 5(-3cos(2π((1(-4((1x + 3)/(-2x - 2))^2 - 5((1x + 3)/(-2x - 2)) + 3))/(-2(-4((1x + 3)/(-2x - 2))^2 - 5((1x... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(-4(-3((3(4x^2 + 1) + 3)/(2(4x^2 + 1))) + 5)^2 + 2(-3((3(4x^2 + 1) + 3)/(2(4x^2 + 1))) + 5) + 1) + 3)/(-3(-4(-3((3(4x^2 + 1) + 3)/(2(4x^2 + 1))) + 5)^2 + 2(-3((3(4x^2 + 1) + 3)/(2(4x^2 + 1))) + 5) + 1) + 2) and g(x) = -4(2((-3(2|... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-2(1|(-1(5x - 4) + 2) - 2| - 1) - 3)/(2(1|(-1(5x - 4) + 2) - 2| - 1) - 1))^2 - 2 and g(x) = 2|(2(5(2x^2 + 3x + 4) + 1) + 2) - 1| + 3 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 3sin(1π(2(4((-1(5x^2 + 5x - 5) - 3)/(3(5x^2 + 5x - 5) + 3)) - 1)^2 - 4(4((-1(5x^2 + 5x - 5) - 3)/(3(5x^2 + 5x - 5) + 3)) - 1) - 4) - 1) = 3((1(-4(1e^(-2x) - 1)^2 - 5(1e^(-2x) - 1) - 1))/(2(-4(1e^(-2x) - 1)^2 - 5(1e^(-2x) - 1) - 1) - 3)) + 1 have for x in the in... | 105 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-3cos(2π(-2(1(-5x^2 - 4x + 3) - 1)^2 - 3(1(-5x^2 - 4x + 3) - 1) - 2) - 1) + 2) and g(x) = -1e^(-1((-1(-2|((-1x + 1)/(-3x + 4)) + 3| - 2) - 2)/(3(-2|((-1x + 1)/(-3x + 4)) + 3| - 2)))) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in L... | 762 | MATH |
[
{
"content": "How many solutions does the equation 3((-2((1(-2(-1|x - 3| + 1)^2 - 5(-1|x - 3| + 1) - 1) + 3)/(-3(-2(-1|x - 3| + 1)^2 - 5(-1|x - 3| + 1) - 1) - 3)))/(1((1(-2(-1|x - 3| + 1)^2 - 5(-1|x - 3| + 1) - 1) + 3)/(-3(-2(-1|x - 3| + 1)^2 - 5(-1|x - 3| + 1) - 1) - 3)) + 1)) - 3 = 4(-2|((1(-2|x - 2| + 2) + 3... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3(-2sin(3π(4(-2(-1x + 5) + 5) - 1) + 2) + 1) + 3 = (-1(-4(2|((1x + 1)/(-1x - 2)) + 2|)^2 + 2(2|((1x + 1)/(-1x - 2)) + 2|) + 4) + 1)/(1(-4(2|((1x + 1)/(-1x - 2)) + 2|)^2 + 2(2|((1x + 1)/(-1x - 2)) + 2|) + 4)) have for x in the interval [-10, 10]?\n\nPresent the ... | 11 | MATH |
[
{
"content": "How many solutions does the equation 3(5(2|(-3((1x - 1)/(-3x + 4))^2 + 5((1x - 1)/(-3x + 4)) + 2) + 1| + 1) - 5) - 3 = 3(-4(2|((-1x - 2)/(-3x + 1)) - 2|)^2 - 3(2|((-1x - 2)/(-3x + 1)) - 2|) - 4)^2 + 4(-4(2|((-1x - 2)/(-3x + 1)) - 2|)^2 - 3(2|((-1x - 2)/(-3x + 1)) - 2|) - 4) - 1 have for x in the i... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(4((3(-2(3sin(1πx - 2) - 2) - 4) - 1)/(3(-2(3sin(1πx - 2) - 2) - 4) + 1)) + 3)| - 1 and g(x) = (-2((1(-2cos(1π(-2x - 3) + 3) + 1) + 1)/(-3(-2cos(1π(-2x - 3) + 3) + 1) + 4)) + 1)/(-1((1(-2cos(1π(-2x - 3) + 3) + 1) + 1)/(-3(-... | 37 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-2|(-3|(-1sin(2π(1x^2 - 4x - 3)) - 2) + 2| + 3) + 3| - 3)^2 + 2 and g(x) = 4(-5(1((-2x + 2)/(-1x - 2)) + 1) - 5) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((2(2|(2(2cos(2πx - 1) + 3) + 1) - 3| - 1) - 3)/(1(2|(2(2cos(2πx - 1) + 3) + 1) - 3| - 1))) + 3| + 1 and g(x) = 4(-1((-1(-1x) - 2)/(-3(-1x)))^2 - 1((-1(-1x) - 2)/(-3(-1x))) + 4) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answe... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-3sin(1π((-2((2((-3x - 2)/(-3x + 1)) - 2)/(1((-3x - 2)/(-3x + 1)) + 3)) - 2)/(3((2((-3x - 2)/(-3x + 1)) - 2)/(1((-3x - 2)/(-3x + 1)) + 3)) - 2)) - 3) + 3) - 1| - 1 and g(x) = (1(-1(-1|(-1cos(3πx) - 1) + 2| + 2)^2 - 3))/(-3(-1(-1... | 27 | MATH |
[
{
"content": "How many solutions does the equation -3(2(2cos(3π(-3|(-2x^2 - 3x + 3) - 3| + 1) + 1) + 3) - 5) - 1 = (1(-4(-5(-4x^2 - 5x + 4)^2 + 2(-4x^2 - 5x + 4) + 2) + 5) - 2)/(-2(-4(-5(-4x^2 - 5x + 4)^2 + 2(-4x^2 - 5x + 4) + 2) + 5) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex for... | 1627 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(1(1|(2sin(1π((2x - 1)/(-1x - 1))) - 2)| - 2)^2 - 1(1|(2sin(1π((2x - 1)/(-1x - 1))) - 2)| - 2) + 3)^2 - 1(1(1|(2sin(1π((2x - 1)/(-1x - 1))) - 2)| - 2)^2 - 1(1|(2sin(1π((2x - 1)/(-1x - 1))) - 2)| - 2) + 3) + 2 and g(x) = -3(3|(1|(-3sin(2πx + 2) - 2) +... | 46 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(3e^(-1(2|(1((-1x + 1)/(-2x + 1)) + 2) + 3| - 1)) - 3) + 2)/(-3(3e^(-1(2|(1((-1x + 1)/(-2x + 1)) + 2) + 3| - 1)) - 3)) and g(x) = -1sin(1π((-3((-1(5x + 5) + 1)/(-3(5x + 5) - 3)) + 2)/(-1((-1(5x + 5) + 1)/(-3(5x + 5) - 3)) + 3))) - 1 intersect in the ... | 15 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((1(2e^(-1(5((-3x + 1)/(-3x - 3))^2 + 1((-3x + 1)/(-3x - 3)) - 2)) - 1) + 2)/(3(2e^(-1(5((-3x + 1)/(-3x - 3))^2 + 1((-3x + 1)/(-3x - 3)) - 2)) - 1)))| and g(x) = 2|(-3(-4(3x + 1) + 2) + 3) + 1| in the range -10 ≤ x ≤ 10?\n\nPrese... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-5(-1(-3((-2x - 1)/(-3x - 1)) + 1) - 2)^2 - 3(-1(-3((-2x - 1)/(-3x - 1)) + 1) - 2) + 4) - 5 and g(x) = (3(1sin(1π(-3(4x + 3)^2 - 4(4x + 3) - 5) - 1) + 2) - 2)/(-1(1sin(1π(-3(4x + 3)^2 - 4(4x + 3) - 5) - 1) + 2) + 2) intersect in the interval [-10, 10... | 108 | MATH |
[
{
"content": "How many solutions does the equation 5(1(1sin(2π(1|((2x + 2)/(2x - 1))| - 2) - 3))^2 + 5(1sin(2π(1|((2x + 2)/(2x - 1))| - 2) - 3)) + 5)^2 + 4(1(1sin(2π(1|((2x + 2)/(2x - 1))| - 2) - 3))^2 + 5(1sin(2π(1|((2x + 2)/(2x - 1))| - 2) - 3)) + 5) = 1e^(-1(-2|(-1((1x - 2)/(2x + 2))^2 + 2((1x - 2)/(2x + 2))... | 52 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-2|((2(2|((2x + 1)/(-2x)) - 3| - 3) + 1)/(2(2|((2x + 1)/(-2x)) - 3| - 3) - 1)) - 1|) + 2| + 2 and g(x) = 1((-1(-1|(-1|x - 1| - 2) - 3| + 2) + 1)/(-1(-1|(-1|x - 1| - 2) - 3| + 2))) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaT... | 4 | MATH |
[
{
"content": "How many solutions does the equation -4(-5(1(2sin(2π(3|x + 2| - 3)) + 2)^2 + 4(2sin(2π(3|x + 2| - 3)) + 2) - 5) - 5)^2 + 5(-5(1(2sin(2π(3|x + 2| - 3)) + 2)^2 + 4(2sin(2π(3|x + 2| - 3)) + 2) - 5) - 5) + 4 = 3((-1(3cos(2π((3x + 3)/(-2x - 1)) + 1) - 2) - 2)/(3(3cos(2π((3x + 3)/(-2x - 1)) + 1) - 2) + ... | 140 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-3(1|((2(1cos(1πx - 3)) - 3)/(2(1cos(1πx - 3)) + 2))| - 3) + 2) + 1)/(-2(-3(1|((2(1cos(1πx - 3)) - 3)/(2(1cos(1πx - 3)) + 2))| - 3) + 2) - 3) and g(x) = (-1(3(-3|(5x + 3) - 3| + 3) - 2) + 2)/(2(3(-3|(5x + 3) - 3| + 3) - 2) + 1)... | 22 | MATH |
[
{
"content": "How many solutions does the equation 5(-3sin(3π(-2|((1(-3|x + 1| + 3) + 3)/(2(-3|x + 1| + 3) - 1)) + 1| + 2) - 2) - 2) = 2(-3|(-2e^(-2(-1x^2 + 5x - 5)) - 2) + 3|) + 4 have for x 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) = -2(-1(-1(2(-2x)^2 - 5(-2x) - 1)^2 - 1(2(-2x)^2 - 5(-2x) - 1))^2 + 5(-1(2(-2x)^2 - 5(-2x) - 1)^2 - 1(2(-2x)^2 - 5(-2x) - 1)) - 1) - 1 and g(x) = (-1(-1(-1(-4x^2 + 2x)^2 + 4(-4x^2 + 2x) - 1) + 4))/(2(-1(-1(-4x^2 + 2x)^2 + 4(-4x^2 + 2x) - 1) + 4) - 1) int... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(4(-2|(-4(-2x^2 - 5x + 5)^2 - 1(-2x^2 - 5x + 5) + 5)|) + 1) - 5 and g(x) = -2|((3((-3(-1x + 4) + 3)/(-2(-1x + 4) + 3)) + 3)/(-3((-3(-1x + 4) + 3)/(-2(-1x + 4) + 3)) + 1))| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in L... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-2|(5(1(-2|x - 1| + 2)^2 + 2(-2|x - 1| + 2))^2 + 2(1(-2|x - 1| + 2)^2 + 2(-2|x - 1| + 2)) + 1) - 3|) + 3 and g(x) = 2(1e^(-1(3|((-1x)/(2x + 2)) - 1| + 2)))^2 + 1(1e^(-1(3|((-1x)/(2x + 2)) - 1| + 2))) - 5 where -10 ≤ x ≤ 10.... | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2|(-2|((-3(3|x + 2| - 3) - 3)/(1(3|x + 2| - 3)))| - 2) + 2|) + 5 and g(x) = 1(-1(1((3x - 2)/(3x - 1))^2 - 5((3x - 2)/(3x - 1)) + 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 6 | MATH |
[
{
"content": "How many solutions does the equation -2(1(1sin(2π(3|(3|x| + 1) - 2| - 2) + 1) + 2)^2 - 3) + 4 = 2((1(-2(-2|x + 3| + 3) + 5) - 1)/(-1(-2(-2|x + 3| + 3) + 5) - 3))^2 - 5((1(-2(-2|x + 3| + 3) + 5) - 1)/(-1(-2(-2|x + 3| + 3) + 5) - 3)) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in... | 359 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3((2(-3e^(2(1(-2x^2 - 2x - 1) - 4)) + 1) - 2)/(-1(-3e^(2(1(-2x^2 - 2x - 1) - 4)) + 1) + 2))^2 + 5((2(-3e^(2(1(-2x^2 - 2x - 1) - 4)) + 1) - 2)/(-1(-3e^(2(1(-2x^2 - 2x - 1) - 4)) + 1) + 2)) + 1 and g(x) = -3|(1|(-1|(3x - 5) + ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|((-1((-2((2(5x^2 - 3x + 2) + 3)/(1(5x^2 - 3x + 2))) - 3)/(-2((2(5x^2 - 3x + 2) + 3)/(1(5x^2 - 3x + 2))) + 1)) - 3)/(1((-2((2(5x^2 - 3x + 2) + 3)/(1(5x^2 - 3x + 2))) - 3)/(-2((2(5x^2 - 3x + 2) + 3)/(1(5x^2 - 3x + 2))) + 1)) + 3))... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(3|(1|(4((-2x + 3)/(-3x + 1))^2 + 2((-2x + 3)/(-3x + 1)) + 2) - 2| - 1) + 3| - 3) - 1)/(2(3|(1|(4((-2x + 3)/(-3x + 1))^2 + 2((-2x + 3)/(-3x + 1)) + 2) - 2| - 1) + 3| - 3) - 3) and g(x) = -4(3cos(3π((1(-5x - 3) + 2)/(-3(-5x ... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(1(-3(-1sin(2π(-3x - 3) - 3) + 1))^2 + 4(-3(-1sin(2π(-3x - 3) - 3) + 1)) + 1) + 4 and g(x) = 3((2((-1(1sin(2πx + 3) + 2) + 1)/(3(1sin(2πx + 3) + 2) + 3)))/(1((-1(1sin(2πx + 3) + 2) + 1)/(3(1sin(2πx + 3) + 2) + 3))))^2 - 1((... | 240 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-2(-5(-2|(-2x^2 - 5x)| + 2) - 1)^2 + 1(-5(-2|(-2x^2 - 5x)| + 2) - 1) + 4) - 3 and g(x) = (-3((1((-2((-2x - 2)/(-3x - 3)))/(-1((-2x - 2)/(-3x - 3)))) + 2)/(3((-2((-2x - 2)/(-3x - 3)))/(-1((-2x - 2)/(-3x - 3)))) - 2)))/(-1((1((-2(... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(1(2e^(-1(1|(-1|x + 3| - 1)|)) - 2) - 4) and g(x) = -1|((-3((2(1|x + 2| - 2) + 3)/(3(1|x + 2| - 2))) + 3)/(1((2(1|x + 2| - 2) + 3)/(3(1|x + 2| - 2))) + 2)) - 2| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format:... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(2sin(1π((-1(1|(-3x^2 + 3x + 1) + 1|) + 2)/(-2(1|(-3x^2 + 3x + 1) + 1|) - 1)) + 2) + 2)^2 + 3(2sin(1π((-1(1|(-3x^2 + 3x + 1) + 1|) + 2)/(-2(1|(-3x^2 + 3x + 1) + 1|) - 1)) + 2) + 2) - 3 and g(x) = (-2(1(2|(-2sin(3πx + 1) + 3)... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1sin(1π(-1(-3(1(2|x + 2| + 3) + 1)^2 - 3(1(2|x + 2| + 3) + 1) + 3)^2) - 1) + 3 and g(x) = 3((3(-1|(-2e^(-1x) + 1) + 3| + 3) - 3)/(2(-1|(-2e^(-1x) + 1) + 3| + 3) - 3))^2 - 1((3(-1|(-2e^(-1x) + 1) + 3| + 3) - 3)/(2(-1|(-2e^(-1... | 35 | MATH |
[
{
"content": "How many solutions does the equation -1(-1|((-2(-2sin(2π(5x - 5) - 3) - 1) + 2)/(-3(-2sin(2π(5x - 5) - 3) - 1))) - 1| + 2) + 3 = 3(2e^(1(-5(-1x - 4)^2 + 1(-1x - 4) + 1)) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 293 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-1cos(3π(3(3((-1x - 3)/(-2x - 3)) - 4)^2 - 5(3((-1x - 3)/(-2x - 3)) - 4) - 2) + 2) - 2)^2 + 3(-1cos(3π(3(3((-1x - 3)/(-2x - 3)) - 4)^2 - 5(3((-1x - 3)/(-2x - 3)) - 4) - 2) + 2) - 2) + 5 and g(x) = 1((-2(-3|((1x - 2)/(3x - 1))| - 3))/(1(-3|((1x - 2)/... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(2(-3(3(-3e^(2x) + 2)^2 - 2)^2 + 4(3(-3e^(2x) + 2)^2 - 2) + 3) - 2) - 2| + 3 and g(x) = 2sin(3π(2(-1|((3x + 2)/(-1x)) + 1| + 2) - 2)) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(1|(1(-2(-5x - 1) + 2)^2 - 2(-2(-5x - 1) + 2) - 4) + 1| - 1))/(-2(1|(1(-2(-5x - 1) + 2)^2 - 2(-2(-5x - 1) + 2) - 4) + 1| - 1)) and g(x) = 1|(-1|(-3cos(1π((-1x + 2)/(-1x - 3)) + 2) + 2) + 3| + 3)| - 3 intersect in the interval [-10, 10].\n\nPresent th... | 85 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(2π(5(2|((-2(2x^2 + 4x + 2) + 1)/(-1(2x^2 + 4x + 2) - 2)) - 1| + 1)^2 - 1) - 2) + 2 and g(x) = 5((-1(1|(1|x - 3| + 2) + 2| + 3) - 3)/(3(1|(1|x - 3| + 2) + 2| + 3) + 3)) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in La... | 361 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(5(-4(-1(3|x + 3| - 3)^2 - 4(3|x + 3| - 3) - 2) + 1) + 1)^2 + 2(5(-4(-1(3|x + 3| - 3)^2 - 4(3|x + 3| - 3) - 2) + 1) + 1) and g(x) = 1cos(1π((-1(2(-3x^2 + 2x + 4)^2 - 4(-3x^2 + 2x + 4) + 2) + 2)/(-1(2(-3x^2 + 2x + 4)^2 - 4(-3x^2 + 2x + 4) + 2) - 3)) + ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(5(2|(5(2x + 5)^2 - 1(2x + 5) + 4) + 1| + 2)^2 - 3(2|(5(2x + 5)^2 - 1(2x + 5) + 4) + 1| + 2) + 4) + 3)/(1(5(2|(5(2x + 5)^2 - 1(2x + 5) + 4) + 1| + 2)^2 - 3(2|(5(2x + 5)^2 - 1(2x + 5) + 4) + 1| + 2) + 4) - 2) and g(x) = -2c... | 32 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-2|(-1|(-5(2x^2 + 3x - 4) + 5) - 1| + 2) - 2| + 1) - 3| and g(x) = -2(4((2((2x + 2)/(-3x - 2)))/(-3((2x + 2)/(-3x - 2)) - 1))^2 + 5((2((2x + 2)/(-3x - 2)))/(-3((2x + 2)/(-3x - 2)) - 1)) + 4) + 5 where -10 ≤ x ≤ 10.\n\nPre... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(1π(-2|(-1|(2((-1x - 2)/(1x + 2))^2 - 2((-1x - 2)/(1x + 2)) - 1) + 3| - 1) + 3| + 1) - 1) + 2 and g(x) = 1(-2cos(3π(5(-3|x - 1|)^2 + 4(-3|x - 1|) + 1) - 3) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxe... | 1162 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-3sin(2π(-1|(-5(-5x^2 + 4)) + 1| + 2) - 2)) + 1)/(-1(-3sin(2π(-1|(-5(-5x^2 + 4)) + 1| + 2) - 2)) - 1) and g(x) = 2cos(2π(1(-3(-1x - 1) - 1)^2 - 3(-3(-1x - 1) - 1) + 5) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex for... | 524 | MATH |
[
{
"content": "How many solutions does the equation -2(3|((-2((-3((-3x + 3)/(-2x + 1)) - 1)/(-1((-3x + 3)/(-2x + 1)) + 2)) + 3)/(-2((-3((-3x + 3)/(-2x + 1)) - 1)/(-1((-3x + 3)/(-2x + 1)) + 2)))) - 1| - 1) + 2 = 4((3(-2sin(2π(-1x^2 - 4x - 2) - 2) - 3))/(2(-2sin(2π(-1x^2 - 4x - 2) - 2) - 3) - 1)) - 5 have for x in... | 6 | MATH |
[
{
"content": "How many solutions does the equation (-3(5((-2(-3sin(2π(1x + 3)) - 2) + 1)/(2(-3sin(2π(1x + 3)) - 2) - 1)) + 1))/(2(5((-2(-3sin(2π(1x + 3)) - 2) + 1)/(2(-3sin(2π(1x + 3)) - 2) - 1)) + 1) - 1) = 5(2|(-5(1x - 4)^2 + 5)|) - 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) = 2|((-3(-2|(1((-2x - 3)/(-1x + 2))^2 + 1((-2x - 3)/(-1x + 2)) - 4) - 2| - 1) + 2)/(-1(-2|(1((-2x - 3)/(-1x + 2))^2 + 1((-2x - 3)/(-1x + 2)) - 4) - 2| - 1))) - 2| - 1 and g(x) = -2sin(2π(-3(2|(1|x + 1| - 3) - 1| + 1)^2 - 3(2|(1|x + 1... | 1767 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1((-2(-2(-1(-1|x| + 2) + 5)^2 + 5(-1(-1|x| + 2) + 5) + 1) + 1)/(-2(-2(-1(-1|x| + 2) + 5)^2 + 5(-1(-1|x| + 2) + 5) + 1))))/(-2((-2(-2(-1(-1|x| + 2) + 5)^2 + 5(-1(-1|x| + 2) + 5) + 1) + 1)/(-2(-2(-1(-1|x| + 2) + 5)^2 + 5(-1(-... | 673 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-2|(2|(-3|((1x + 1)/(-3x - 2)) + 2| + 3) - 1| - 1) - 2| - 2) - 3)/(-1(-2|(2|(-3|((1x + 1)/(-3x - 2)) + 2| + 3) - 1| - 1) - 2| - 2) - 3) and g(x) = 1((-2(-2(2cos(1πx - 1)) - 4) + 3)/(-1(-2(2cos(1πx - 1)) - 4) - 2)) - 4 intersect in the interval [-10,... | 5 | MATH |
[
{
"content": "How many solutions does the equation -2((-1(-3(-1|(-4x - 2) + 2| - 1)^2))/(-3(-3(-1|(-4x - 2) + 2| - 1)^2) + 1))^2 - 3((-1(-3(-1|(-4x - 2) + 2| - 1)^2))/(-3(-3(-1|(-4x - 2) + 2| - 1)^2) + 1)) + 1 = -1(4(-4(-3sin(3πx - 1) - 1) - 5)^2 + 5(-4(-3sin(3πx - 1) - 1) - 5))^2 - 3(4(-4(-3sin(3πx - 1) - 1) -... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(1cos(1π(-1(-5(3x^2 + 3x + 5) + 3) + 5) + 2) - 3))/(-3(1cos(1π(-1(-5(3x^2 + 3x + 5) + 3) + 5) + 2) - 3) - 2) and g(x) = -3|(2((-2(-3|x - 2| + 2) + 2)/(1(-3|x - 2| + 2) - 2)) + 3) - 2| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(-3cos(3π(1(1(5x^2 - 3)^2 + 3(5x^2 - 3) + 1)) - 2)) + 3| + 3 and g(x) = -2((2(-2|((-3x)/(-2x)) + 3| + 1))/(-3(-2|((-3x)/(-2x)) + 3| + 1) - 1)) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 917 | MATH |
[
{
"content": "How many solutions does the equation 1cos(3π(3|(-2((-3(5x^2 - 5x - 4) + 2)/(-2(5x^2 - 5x - 4)))^2 + 3((-3(5x^2 - 5x - 4) + 2)/(-2(5x^2 - 5x - 4))) - 2) + 3| + 2) + 3) + 2 = 5(-3cos(1π((-1(1x^2 + 4x + 1) + 3)/(1(1x^2 + 4x + 1) - 3)) + 1) - 3)^2 + 5(-3cos(1π((-1(1x^2 + 4x + 1) + 3)/(1(1x^2 + 4x + 1)... | 86 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-2(5(-2(-4x - 4) + 2) - 4)^2 + 5(5(-2(-4x - 4) + 2) - 4) + 5) + 5 and g(x) = 1(5((-2(5x - 5) + 2)/(2(5x - 5) + 1))^2 + 2)^2 - 5(5((-2(5x - 5) + 2)/(2(5x - 5) + 1))^2 + 2) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3cos(2π(-1|(-2(2(2x + 5)^2 - 2(2x + 5) - 3)^2 + 5) - 2|) - 2) + 3 = 1((-1(-1e^(-1(-5x + 4)) - 3))/(-1(-1e^(-1(-5x + 4)) - 3) - 3))^2 - 4((-1(-1e^(-1(-5x + 4)) - 3))/(-1(-1e^(-1(-5x + 4)) - 3) - 3)) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer... | 91 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(3π((-1(-5((-1(2|x - 3| + 3) - 2)/(3(2|x - 3| + 3) + 2)) + 4) - 1)/(2(-5((-1(2|x - 3| + 3) - 2)/(3(2|x - 3| + 3) + 2)) + 4) + 3)) + 1) and g(x) = 3((1(2(2|x - 3| - 2) - 2))/(-2(2(2|x - 3| - 2) - 2) - 2))^2 + 5((1(2(2|x - ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-3(2sin(2π(5(-4x)) - 3) + 1) - 5)^2 + 1(-3(2sin(2π(5(-4x)) - 3) + 1) - 5) + 5 and g(x) = -1(-1|((-1((-3x - 3)/(-3x - 2)) + 3)/(2((-3x - 3)/(-3x - 2)) - 3)) - 3| - 3)^2 + 1(-1|((-1((-3x - 3)/(-3x - 2)) + 3)/(2((-3x - 3)/(-3... | 793 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1e^(2(1((-1((-3((-1x - 1)/(1x)) - 3)/(-3((-1x - 1)/(1x)) - 3)) - 3)/(-3((-3((-1x - 1)/(1x)) - 3)/(-3((-1x - 1)/(1x)) - 3)) - 1))^2 - 1((-1((-3((-1x - 1)/(1x)) - 3)/(-3((-1x - 1)/(1x)) - 3)) - 3)/(-3((-3((-1x - 1)/(1x)) - 3)/... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(3cos(1π(5((3(5x^2 - 1x + 2))/(3(5x^2 - 1x + 2) + 3))^2 + 1((3(5x^2 - 1x + 2))/(3(5x^2 - 1x + 2) + 3)) - 2) + 2) - 3)^2 - 3(3cos(1π(5((3(5x^2 - 1x + 2))/(3(5x^2 - 1x + 2) + 3))^2 + 1((3(5x^2 - 1x + 2))/(3(5x^2 - 1x + 2) + 3)) - 2) + 2) - 3) - 2 and g... | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((-2(-2(-4(-3x + 3) - 4)^2 + 2(-4(-3x + 3) - 4) + 5) - 1)/(2(-2(-4(-3x + 3) - 4)^2 + 2(-4(-3x + 3) - 4) + 5) + 2)) - 1 and g(x) = 5((-1(-1(-1x - 1)^2 + 3(-1x - 1) + 4) - 3)/(-3(-1(-1x - 1)^2 + 3(-1x - 1) + 4) + 1)) - 1 where... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(2((2(5(4x - 3)^2 - 1(4x - 3) + 3) + 3)/(3(5(4x - 3)^2 - 1(4x - 3) + 3) + 1))^2 + 3((2(5(4x - 3)^2 - 1(4x - 3) + 3) + 3)/(3(5(4x - 3)^2 - 1(4x - 3) + 3) + 1)) - 5) + 2 and g(x) = -3(-1|(2(1cos(3πx) - 2)^2 + 3(1cos(3πx) - 2) - 5) +... | 95 | MATH |
[
{
"content": "How many solutions does the equation (1(2|(1(2(-1|x|)^2 + 1(-1|x|) + 1) - 5) - 1| - 3) - 3)/(1(2|(1(2(-1|x|)^2 + 1(-1|x|) + 1) - 5) - 1| - 3) - 2) = 4(-4(2|((-2x - 1)/(1x + 2)) - 1| - 2)^2 + 2(2|((-2x - 1)/(1x + 2)) - 1| - 2) + 1)^2 + 2(-4(2|((-2x - 1)/(1x + 2)) - 1| - 2)^2 + 2(2|((-2x - 1)/(1x + ... | 8 | MATH |
[
{
"content": "How many solutions does the equation 2((2(3|(1(-5x + 5) - 2) - 2| + 1) + 2)/(3(3|(1(-5x + 5) - 2) - 2| + 1))) - 5 = 2((3(-5(2x^2 - 4x - 1)))/(3(-5(2x^2 - 4x - 1)) + 3))^2 - 5((3(-5(2x^2 - 4x - 1)))/(3(-5(2x^2 - 4x - 1)) + 3)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-2|(-1sin(2π(-1(-1x + 4)^2 - 5(-1x + 4) + 5) + 2)) - 1| + 2) + 2 and g(x) = 2|(2|(-1((-3x + 2)/(1x + 1))^2 - 3((-3x + 2)/(1x + 1)) + 5) + 3| - 2) - 3| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed... | 119 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(4(4(-5(-3cos(1πx + 2) - 2)^2 - 2(-3cos(1πx + 2) - 2) - 2)^2 - 5(-5(-3cos(1πx + 2) - 2)^2 - 2(-3cos(1πx + 2) - 2) - 2)) + 1) + 2)/(1(4(4(-5(-3cos(1πx + 2) - 2)^2 - 2(-3cos(1πx + 2) - 2) - 2)^2 - 5(-5(-3cos(1πx + 2) - 2)^2 -... | 307 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(2(-1cos(3π(-3(-3|x - 3| + 2) + 2) + 3) - 3) + 4)^2 - 4(2(-1cos(3π(-3(-3|x - 3| + 2) + 2) + 3) - 3) + 4) + 1 and g(x) = 2cos(2π(-3(2(-2|x + 2|)^2 + 4(-2|x + 2|) + 4)^2 - 1(2(-2|x + 2|)^2 + 4(-2|x + 2|) + 4) + 5) + 2) + 3 intersect in the interval [-1... | 378 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2((3(1|(1|((3x + 2)/(-1x + 2)) - 2| + 1) - 3| - 1) + 1)/(-3(1|(1|((3x + 2)/(-1x + 2)) - 2| + 1) - 3| - 1) + 4)) + 3)/(1((3(1|(1|((3x + 2)/(-1x + 2)) - 2| + 1) - 3| - 1) + 1)/(-3(1|(1|((3x + 2)/(-1x + 2)) - 2| + 1) - 3| - 1)... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-3sin(1π(1|((1(-4x^2 - 2x + 3) - 1)/(3(-4x^2 - 2x + 3) + 2))|) + 2) - 1) - 1 and g(x) = -3cos(2π(-1(3(-4x - 4)^2 - 5(-4x - 4) - 5)^2 - 5) + 2) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 51 | MATH |
[
{
"content": "How many solutions does the equation -1(-4(2((-3(-1|x - 3| - 1) + 2)/(1(-1|x - 3| - 1) - 2)) + 2) - 4) + 3 = 2cos(2π((-3(1|(-4x + 4) + 1|) + 1)/(-2(1|(-4x + 4) + 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": "Determine the number of intersection points between the functions f(x) = -5((2(2(-5(2x + 1) - 4)^2 - 4) + 2)/(1(2(-5(2x + 1) - 4)^2 - 4) - 3))^2 + 2((2(2(-5(2x + 1) - 4)^2 - 4) + 2)/(1(2(-5(2x + 1) - 4)^2 - 4) - 3)) + 2 and g(x) = -5(2|(-2|(-1|x + 1|) - 1| + 2)| + 3) + 1 where -10 ≤ x ≤ 10.\n\nPre... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(1π(-4(3(5((3x - 1)/(2x))^2 + 1((3x - 1)/(2x)) + 4)^2 + 3(5((3x - 1)/(2x))^2 + 1((3x - 1)/(2x)) + 4) - 5) - 3) - 1) + 3 and g(x) = -1|(1|(-5(1x - 4) + 3) + 3| - 2)| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex ... | 123 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-3cos(2π(1(-3|(2|x + 2|) + 2| - 1) + 4) - 3)) - 3| - 3 and g(x) = -1(-3|(-1|((3x - 1)/(1x + 3))| + 3) + 2| + 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 164 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-3|(-3cos(3π(5(3x + 3) - 2) + 1)) - 3| - 2) - 3)/(1(-3|(-3cos(3π(5(3x + 3) - 2) + 1)) - 3| - 2) + 2) and g(x) = 1cos(2π(1|(-2|(-3|x - 3| - 1) - 3| + 3)|) - 3) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format:... | 484 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2((3(-1(-2((3x + 3)/(-1x + 2)) - 2) + 1) - 1)/(1(-1(-2((3x + 3)/(-1x + 2)) - 2) + 1))))/(2((3(-1(-2((3x + 3)/(-1x + 2)) - 2) + 1) - 1)/(1(-1(-2((3x + 3)/(-1x + 2)) - 2) + 1))) + 2) and g(x) = -1cos(1π(1(3(-2x^2 - 4x + 5)^2 -... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|((-1(-1|(2((-2x - 3)/(1x + 1))^2 + 3((-2x - 3)/(1x + 1)) + 5) - 3|) + 2)/(-2(-1|(2((-2x - 3)/(1x + 1))^2 + 3((-2x - 3)/(1x + 1)) + 5) - 3|) - 1)) - 2| + 3 and g(x) = 4(1|(1(3x + 1)^2 + 2(3x + 1) - 3) + 2| + 1)^2 - 5(1|(1(3x + 1)^2 + 2(3x + 1) - 3) +... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(3((3(-5(-3e^(1x) + 2)^2 - 4(-3e^(1x) + 2)) - 1)/(-3(-5(-3e^(1x) + 2)^2 - 4(-3e^(1x) + 2)) - 2))^2 - 3((3(-5(-3e^(1x) + 2)^2 - 4(-3e^(1x) + 2)) - 1)/(-3(-5(-3e^(1x) + 2)^2 - 4(-3e^(1x) + 2)) - 2)) + 2)^2 + 3(3((3(-5(-3e^(1x... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-4(-2(3(1sin(3πx + 1) - 3) - 1)^2 + 2(3(1sin(3πx + 1) - 3) - 1) + 5) - 1) - 3 and g(x) = 1|(-4((2(5x^2 - 5))/(-3(5x^2 - 5) - 3))^2 - 3((2(5x^2 - 5))/(-3(5x^2 - 5) - 3)) + 1) - 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer i... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-3(3|((3(1sin(3π((-1x + 2)/(2x + 1))) + 2) - 2)/(-3(1sin(3π((-1x + 2)/(2x + 1))) + 2))) + 2| - 2))/(1(3|((3(1sin(3π((-1x + 2)/(2x + 1))) + 2) - 2)/(-3(1sin(3π((-1x + 2)/(2x + 1))) + 2))) + 2| - 2) + 3) = (3(-2|(1|(-5x + 5) - 2| + 2) - 3| - 3))/(-3(-2|(1|(-5x +... | 15 | MATH |
[
{
"content": "How many solutions does the equation -1((-1(-1e^(2(1((1x - 2)/(-2x))^2 - 1((1x - 2)/(-2x)) - 2)) + 2) + 3)/(2(-1e^(2(1((1x - 2)/(-2x))^2 - 1((1x - 2)/(-2x)) - 2)) + 2)))^2 + 1((-1(-1e^(2(1((1x - 2)/(-2x))^2 - 1((1x - 2)/(-2x)) - 2)) + 2) + 3)/(2(-1e^(2(1((1x - 2)/(-2x))^2 - 1((1x - 2)/(-2x)) - 2))... | 6 | MATH |
[
{
"content": "How many solutions does the equation -1((-1((1(3|((2x - 1)/(-2x - 1)) - 3| + 3) + 2)/(-1(3|((2x - 1)/(-2x - 1)) - 3| + 3) - 2)) + 2)/(-1((1(3|((2x - 1)/(-2x - 1)) - 3| + 3) + 2)/(-1(3|((2x - 1)/(-2x - 1)) - 3| + 3) - 2))))^2 + 5((-1((1(3|((2x - 1)/(-2x - 1)) - 3| + 3) + 2)/(-1(3|((2x - 1)/(-2x - 1... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|((2((-2(2cos(3π(2|x| - 2) + 3) - 3) + 1)/(-2(2cos(3π(2|x| - 2) + 3) - 3) - 3)) - 3)/(3((-2(2cos(3π(2|x| - 2) + 3) - 3) + 1)/(-2(2cos(3π(2|x| - 2) + 3) - 3) - 3)) - 2))| - 1 and g(x) = -2sin(2π(2(3(-1x^2 + 2x + 3)^2 + 4(-1x^2 + 2... | 1290 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-1((-2(-2|(5x + 1)| + 2) - 2)/(2(-2|(5x + 1)| + 2) - 1)) + 5) - 3| - 2 and g(x) = 2|(1|(2cos(2π((-3x + 3)/(-2x - 1)) - 3)) - 1| + 3) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 51 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-1cos(3π(-3|(2(-2|x| - 1)^2 + 3(-2|x| - 1) + 2) + 1| + 2) - 2) - 2) - 3)/(2(-1cos(3π(-3|(2(-2|x| - 1)^2 + 3(-2|x| - 1) + 2) + 1| + 2) - 2) - 2)) and g(x) = 1(1(-3|(3sin(2πx - 1)) - 1| + 3)^2 - 4(-3|(3sin(2πx - 1)) - 1| + 3) + 2... | 70 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3sin(1π(5(3|(3|((2x - 1)/(2x + 1)) - 1| - 1) - 3| + 1) + 2) - 1) - 3 and g(x) = 1|(-1(-3cos(3π(5x^2 + 1x - 2) + 2) - 3)^2 + 2(-3cos(3π(5x^2 + 1x - 2) + 2) - 3) + 1) + 2| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex... | 220 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-2cos(1π((-2(2((3x + 2)/(-2x - 3))^2 + 2((3x + 2)/(-2x - 3)) + 5) + 3)/(1(2((3x + 2)/(-2x - 3))^2 + 2((3x + 2)/(-2x - 3)) + 5)))) + 3)^2 - 5(-2cos(1π((-2(2((3x + 2)/(-2x - 3))^2 + 2((3x + 2)/(-2x - 3)) + 5) + 3)/(1(2((3x + 2)/(-2... | 54 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(1π(1|(1|(3(-3|x - 2| + 3)^2 + 1(-3|x - 2| + 3) - 4) - 1| + 1)| - 2) + 2) - 1 and g(x) = -3|(-1((1(5x - 4) - 1)/(1(5x - 4) + 3))^2 - 3((1(5x - 4) - 1)/(1(5x - 4) + 3))) - 1| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer ... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(1(2sin(2π(-3(-1x^2 + 4x - 3)^2 + 2(-1x^2 + 4x - 3) + 2) - 3) + 1)^2 + 5(2sin(2π(-3(-1x^2 + 4x - 3)^2 + 2(-1x^2 + 4x - 3) + 2) - 3) + 1) - 3) + 2| + 1 and g(x) = (1(-4(3(2|x + 1| - 1) - 3) - 2) + 2)/(-3(-4(3(2|x + 1| - 1) - 3) -... | 366 | MATH |
[
{
"content": "How many solutions does the equation (3(-2((2(-4(-2e^(1x) - 2)^2 - 2(-2e^(1x) - 2) - 3))/(3(-4(-2e^(1x) - 2)^2 - 2(-2e^(1x) - 2) - 3)))^2 + 3((2(-4(-2e^(1x) - 2)^2 - 2(-2e^(1x) - 2) - 3))/(3(-4(-2e^(1x) - 2)^2 - 2(-2e^(1x) - 2) - 3))) - 2) - 3)/(2(-2((2(-4(-2e^(1x) - 2)^2 - 2(-2e^(1x) - 2) - 3))/(... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(1π(-3|((1(3|(-3x + 2) + 1| + 1) + 1)/(-3(3|(-3x + 2) + 1| + 1) + 4)) - 1| - 1) - 3) and g(x) = -5(-3sin(3π(-3((-1x - 2)/(3x)) + 1) + 1) - 3)^2 + 4(-3sin(3π(-3((-1x - 2)/(3x)) + 1) + 1) - 3) intersect in the interval [-10, 10].\n\nPresent the answe... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(2π(-5(-2|(5(-1x^2 - 2x + 1)^2 - 3(-1x^2 - 2x + 1) - 3) - 1| + 2)^2 - 4(-2|(5(-1x^2 - 2x + 1)^2 - 3(-1x^2 - 2x + 1) - 3) - 1| + 2) - 2) - 3) - 2 and g(x) = -2e^(1((3((3(-3|x + 3| + 3) - 1)/(2(-3|x + 3| + 3) - 3)) + 2)/(-2((3(-3|x + 3| + 3) - 1)/(2(... | 43 | MATH |
[
{
"content": "How many solutions does the equation 3(2|((-1(-4(-2|x + 2| + 1) + 4) + 3)/(1(-4(-2|x + 2| + 1) + 4) - 3)) + 3| - 3)^2 - 1(2|((-1(-4(-2|x + 2| + 1) + 4) + 3)/(1(-4(-2|x + 2| + 1) + 4) - 3)) + 3| - 3) - 4 = -4(-2(-1((1x - 3)/(-3x + 2))^2 + 1((1x - 3)/(-3x + 2)) + 3) + 5)^2 - 1(-2(-1((1x - 3)/(-3x + ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-1(-3((-1(-2|x + 1|) - 2)/(3(-2|x + 1|))) + 4) - 2) + 5 and g(x) = 2sin(2π(-1(2|(-1x^2 - 5) - 1| - 3) + 1)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 774 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-2(2|((-3(1|x + 1| - 1) + 2)/(2(1|x + 1| - 1) - 3)) - 3| - 1) + 4) + 1)/(-2(-2(2|((-3(1|x + 1| - 1) + 2)/(2(1|x + 1| - 1) - 3)) - 3| - 1) + 4) - 1) and g(x) = (-1((-1((-2((1x)/(-3x - 3)) + 1)/(-1((1x)/(-3x - 3)) + 2)) + 3)/(3((-2((1x)/(-3x - 3)) + 1... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4((1((3(3|(-3x + 5) - 2| + 1))/(-1(3|(-3x + 5) - 2| + 1) + 3)) - 3)/(1((3(3|(-3x + 5) - 2| + 1))/(-1(3|(-3x + 5) - 2| + 1) + 3)) + 3)) + 4 and g(x) = (-2(1(3cos(2π(4x - 5) - 2) - 3) + 2) - 3)/(2(1(3cos(2π(4x - 5) - 2) - 3) + 2) + 2) intersect in the i... | 7 | MATH |
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