[User]
How many points of intersection exist between the functions f(x) = -3((2(-1sin(1π(4x) - 1) + 1) + 1)/(3(-1sin(1π(4x) - 1) + 1) + 3))^2 - 4((2(-1sin(1π(4x) - 1) + 1) + 1)/(3(-1sin(1π(4x) - 1) + 1) + 3)) + 2 and g(x) = -1cos(2π(-2|(3x^2 + 5x - 4) - 3| - 3) + 3) - 2 in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}

dataset: MATH


[User]
How many solutions does the equation (-2(2cos(1π(5(1|x - 2| - 2) - 3) - 2) - 2) + 3)/(-2(2cos(1π(5(1|x - 2| - 2) - 3) - 2) - 2) + 3) = 2(-3|(-3e^(-1x) + 3) + 3| + 1) + 4 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = 2|(-2sin(1πx + 2) + 1) - 2| + 3 and g(x) = 3|x + 2| - 3 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation -5(5x)^2 + 3(5x) + 5 = -5x^2 + 5x - 3 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation -2|((1(1x^2 + 3x + 4) + 3)/(3(1x^2 + 3x + 4) - 1)) - 1| - 1 = -1|((2x - 1)/(-1x + 2)) - 3| - 1 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = 2|(-3(-2((1x - 1)/(1x)) - 5)^2 - 4(-2((1x - 1)/(1x)) - 5) - 1) - 3| - 1 and g(x) = -5((-1(-2cos(2πx) + 1) + 3)/(2(-2cos(2πx) + 1) - 3)) in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation (1(1(2|x| - 3)^2 + 1) + 3)/(-3(1(2|x| - 3)^2 + 1) + 4) = -3cos(1π(-1x^2 + 5x - 2) + 2) - 1 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -1(3sin(3π(1|(5x) + 3| - 2) - 1))^2 + 5(3sin(3π(1|(5x) + 3| - 2) - 1)) - 5 and g(x) = 2|(-4((2x + 2)/(-2x - 1)) - 4) - 1| + 1 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -3sin(3π(1(-3|x| + 3)) + 2) + 2 and g(x) = (-2(-2|x + 2| - 1) + 1)/(2(-2|x + 2| - 1) + 3) intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = (-2(-3x) + 2)/(-1(-3x) + 3) and g(x) = (-1x - 1)/(-1x + 2) in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = -3|(-4(-3cos(1πx) + 2) - 1) + 2| + 3 and g(x) = 3((3x + 1)/(-3x - 2)) - 1 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = -3cos(1π(4(2x^2 - 4x + 4) - 4)) + 1 and g(x) = -1sin(2π((-3x - 3)/(-3x + 4))) + 1 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation -2sin(3π(-1x^2 - 2x + 1) - 3) - 1 = -2|x + 1| - 2 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = -4((1x + 3)/(3x - 2))^2 + 4((1x + 3)/(3x - 2)) - 4 and g(x) = -1x^2 + 2x - 2 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation (3(-3|(-3e^(2(1x + 2)) - 3)| - 1) - 1)/(-1(-3|(-3e^(2(1x + 2)) - 3)| - 1)) = (1(-5(1x^2 - 5x + 3) + 1) + 1)/(2(-5(1x^2 - 5x + 3) + 1)) have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = 4(5(-2sin(1πx + 2) - 1) + 4)^2 - 1(5(-2sin(1πx + 2) - 1) + 4) - 2 and g(x) = 1(1x + 2)^2 - 5(1x + 2) + 4 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -3(2|(1(-2sin(1πx + 1) - 2)^2 - 4) + 1|) + 3 and g(x) = 5(3(3e^(-2x) + 1) - 3)^2 + 3(3(3e^(-2x) + 1) - 3) - 5 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = -2e^(-1(5((-3(-3x + 5) + 1)/(2(-3x + 5) + 2))^2 + 3((-3(-3x + 5) + 1)/(2(-3x + 5) + 2)) - 4)) and g(x) = (-1(1sin(3π(-1x - 4) - 1) - 3) - 3)/(1(1sin(3π(-1x - 4) - 1) - 3) - 3) where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = 1|(3((-2x + 1)/(-3x - 1))) + 1| and g(x) = 1sin(2π((-3x + 2)/(-3x + 4)) - 1) in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = 2sin(3π(-1|(3(1x^2 + 3x - 1) + 2) + 3|) - 1) - 3 and g(x) = -1(-2|(-1cos(1πx + 3) + 1) + 3| + 2) - 5 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -3|((-2(5(3x^2 + 3x - 3) - 2) + 1)/(3(5(3x^2 + 3x - 3) - 2))) + 1| - 3 and g(x) = -3cos(3π(2|(5x) - 2| + 2) - 3) - 1 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = (1(1(1x^2 - 5x - 2)) + 3)/(2(1(1x^2 - 5x - 2)) + 1) and g(x) = 2sin(3π(3x - 3)) - 1 in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = -3(1|x + 2| - 2) - 1 and g(x) = 2|x + 3| in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = 3(3|((2x + 2)/(-3x + 4)) - 1| - 2) - 4 and g(x) = -2|(-5x^2 + 5x - 4) + 3| + 2 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = 2|(4(2((1x + 3)/(1x + 2)) - 5)^2 - 3(2((1x + 3)/(1x + 2)) - 5) + 4) - 3| - 3 and g(x) = 4(4((-2x - 3)/(1x + 2)) - 2)^2 + 1 in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = (2(-3(-1|x + 1| - 2)^2 - 5(-1|x + 1| - 2) + 3))/(3(-3(-1|x + 1| - 2)^2 - 5(-1|x + 1| - 2) + 3) - 2) and g(x) = -1cos(1π(1x^2 - 2x - 3)) + 1 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = (-1(3|(2x + 5) + 1| + 2))/(2(3|(2x + 5) + 1| + 2) + 3) and g(x) = -1sin(2π(-3|x - 3| + 2) + 2) - 1 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -5(1|(1((-3x + 2)/(2x + 1))^2 + 5((-3x + 2)/(2x + 1)) + 4) - 1| - 2) + 5 and g(x) = (1(-2e^(-1(3x - 5)) - 1) - 3)/(-2(-2e^(-1(3x - 5)) - 1) - 1) intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation 3cos(2π((-2x - 1)/(1x)) - 3) + 3 = (1x - 2)/(2x - 3) have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation (-1(1(-3(2|x + 3| - 2) - 1)^2 + 5(-3(2|x + 3| - 2) - 1)) - 2)/(-2(1(-3(2|x + 3| - 2) - 1)^2 + 5(-3(2|x + 3| - 2) - 1)) + 3) = (-1(-1(5x^2 + 3x - 1) - 5) + 3)/(3(-1(5x^2 + 3x - 1) - 5) + 2) have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = -2(4x + 3)^2 - 3(4x + 3) and g(x) = 5x - 5 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -3|(-2sin(1πx) + 3) - 3| + 2 and g(x) = (1x - 1)/(-2x - 3) intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = (-2(1(2|(-2x - 2) - 1| - 3)^2 - 4(2|(-2x - 2) - 1| - 3) + 2) + 1)/(2(1(2|(-2x - 2) - 1| - 3)^2 - 4(2|(-2x - 2) - 1| - 3) + 2) - 1) and g(x) = -2(-3|(-2cos(2πx + 2) - 2)| + 2) - 2 in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -5(1cos(1πx - 2) + 2) + 4 and g(x) = 2|x + 2| - 2 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = 2sin(1π(1x^2 + 1x + 4) - 2) - 3 and g(x) = 2x^2 - 3x - 1 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = 3(1sin(2πx + 2) + 2) + 1 and g(x) = 3x^2 - 4x + 1 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = (1(4(3x - 5)^2 - 1(3x - 5)))/(-1(4(3x - 5)^2 - 1(3x - 5)) - 2) and g(x) = (-2(-4x - 3) - 3)/(2(-4x - 3) - 1) in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation 3sin(3π(1|(-2x^2 - 5x + 2) - 2|) + 3) - 3 = -1((-1x - 1)/(2x - 3))^2 - 4((-1x - 1)/(2x - 3)) + 1 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = 2|(-4((-3x + 2)/(-2x - 1))^2 + 1((-3x + 2)/(-2x - 1)) + 5) + 3| + 3 and g(x) = 5(1|x + 3| + 3) - 3 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation -5(-1e^(-1x) + 3)^2 - 3(-1e^(-1x) + 3) + 5 = -3x^2 - 3 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = (2(-3|(2e^(1(-4x + 5)) - 1) + 3| - 3) + 2)/(-3(-3|(2e^(1(-4x + 5)) - 1) + 3| - 3) + 4) and g(x) = 4(3((-2x + 2)/(-2x - 3))^2 - 2((-2x + 2)/(-2x - 3)) - 5) intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -2cos(1π(5((1x - 2)/(1x))^2 + 1((1x - 2)/(1x)) + 5) + 3) + 2 and g(x) = 1|(2cos(1πx - 3) - 1) - 2| + 1 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -1|(-1(-4((-2x + 3)/(-3x - 1)) + 3)^2 - 2(-4((-2x + 3)/(-3x - 1)) + 3) - 4) + 1| - 2 and g(x) = -3((2(-2e^(-2x)) + 3)/(-1(-2e^(-2x))))^2 + 5((2(-2e^(-2x)) + 3)/(-1(-2e^(-2x)))) + 1 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = -1(3x^2 - 4x + 5) + 3 and g(x) = -2|x + 1| - 1 in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = (-1(3e^(-2x) - 1) - 2)/(-1(3e^(-2x) - 1) + 3) and g(x) = (2x - 1)/(2x) where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = -4(-1|(3x + 1) - 2| + 3)^2 - 3(-1|(3x + 1) - 2| + 3) + 3 and g(x) = -3|((1x - 1)/(2x + 2))| - 1 in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = -2(2(1|x - 1| - 3) - 2)^2 - 3(2(1|x - 1| - 3) - 2) + 1 and g(x) = (2(-3|x + 3| - 1))/(2(-3|x + 3| - 1)) in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -1e^(2((-3x)/(-3x - 2))) - 2 and g(x) = -1|x - 3| - 3 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation (3(5(1|x - 2| - 2) + 5) + 1)/(1(5(1|x - 2| - 2) + 5)) = 3|(-1x^2 - 5x - 3) - 3| - 1 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation 2|(1(1(2cos(3πx + 1) - 3)^2 + 3(2cos(3πx + 1) - 3) + 1) + 2) - 1| + 3 = 2|(2((-2x + 3)/(1x + 2))^2 - 1((-2x + 3)/(1x + 2)) + 2) - 3| - 3 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = (-1(-3sin(3πx - 3) + 2) - 1)/(-1(-3sin(3πx - 3) + 2) - 3) and g(x) = -5x^2 + 2x + 1 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation -2|(1((3(-4x - 3) - 1)/(2(-4x - 3) - 1))^2 + 2((3(-4x - 3) - 1)/(2(-4x - 3) - 1)) - 5) + 3| + 2 = -4(-2|((-1x + 3)/(-1x)) + 3| - 3)^2 + 2(-2|((-1x + 3)/(-1x)) + 3| - 3) + 3 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -3cos(3π(1((-1(-1x^2 + 1) + 1)/(2(-1x^2 + 1) + 2)) - 4) + 2) - 2 and g(x) = (2(1|(-1e^(-1x) - 1) - 1|) + 2)/(3(1|(-1e^(-1x) - 1) - 1|) + 3) intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = -2(-5((-2(2|x - 3| + 3) + 2)/(-1(2|x - 3| + 3) - 3))^2 - 1((-2(2|x - 3| + 3) + 2)/(-1(2|x - 3| + 3) - 3)) - 5) + 2 and g(x) = 3(3((2x)/(3x + 1))^2 - 2((2x)/(3x + 1)) + 1) in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = (1(2|(-4(2x - 2)^2 + 4) + 2| + 2) + 2)/(3(2|(-4(2x - 2)^2 + 4) + 2| + 2) + 1) and g(x) = 3(1|(-1x - 3) - 2| - 1)^2 + 5(1|(-1x - 3) - 2| - 1) + 2 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation -1(1sin(2π((-1(-4x^2 - 2x - 1) + 2)/(2(-4x^2 - 2x - 1) + 3)) + 2) - 1) - 2 = 5(-1sin(2π(-1x^2 - 5x + 3) - 1)) + 1 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = (1(-2(-4x^2 + 4x + 1) - 1) + 2)/(2(-2(-4x^2 + 4x + 1) - 1) + 2) and g(x) = 3(2cos(1πx) + 3) - 3 intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = 1|(-3sin(1πx - 1) + 2) + 2| + 2 and g(x) = 1|x| + 3 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation (2(-2|(-3(1x^2 + 1x - 3) - 2) + 1| - 3) + 2)/(-3(-2|(-3(1x^2 + 1x - 3) - 2) + 1| - 3) + 4) = -1((1(3|x + 1|) - 1)/(-3(3|x + 1|) + 1))^2 - 3((1(3|x + 1|) - 1)/(-3(3|x + 1|) + 1)) + 2 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation -2|(-3(-5x^2 - 2x + 4) - 2) + 1| + 3 = (-3(3|x - 1| + 3) - 1)/(-2(3|x - 1| + 3) + 3) have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation 1e^(2(-3|x - 2| + 3)) - 3 = (-2x + 3)/(-1x + 2) have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = -3(-3|((-2x)/(-2x + 3)) - 3| + 2) - 1 and g(x) = (-1(-3cos(2πx + 2) + 2) + 2)/(-1(-3cos(2πx + 2) + 2) + 3) in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation -1e^(-1((3(-1(3x^2 + 3x - 5) + 5) + 2)/(2(-1(3x^2 + 3x - 5) + 5)))) + 3 = (3(-3(1sin(2πx + 1) + 1)^2 + 5(1sin(2πx + 1) + 1) - 2) - 3)/(-3(-3(1sin(2πx + 1) + 1)^2 + 5(1sin(2πx + 1) + 1) - 2) - 3) have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = -1|((-3x - 2)/(2x + 1)) - 3| + 2 and g(x) = -3|x + 2| + 1 in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = 1|(1e^(1x)) + 3| - 1 and g(x) = 1x^2 + 4x - 2 where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = 2((2x - 3)/(3x))^2 + 5((2x - 3)/(3x)) - 5 and g(x) = 5x^2 + 1x - 4 in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation 1e^(-2(-2(1x - 4)^2 + 4(1x - 4) + 4)) - 3 = 3(2|x - 2| + 3)^2 + 3(2|x - 2| + 3) + 4 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation -1((1(2|(-1x^2 + 1x + 5) - 1| - 1) - 2)/(-3(2|(-1x^2 + 1x + 5) - 1| - 1))) + 3 = -2sin(3π(1(-3|x + 2| - 3) - 4) - 1) - 1 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = (1(-5(3x^2 + 4x - 4) + 5) - 3)/(2(-5(3x^2 + 4x - 4) + 5) - 1) and g(x) = (3(1cos(2πx - 3) + 2))/(1(1cos(2πx - 3) + 2)) where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many solutions does the equation 2(3x^2 - 1x - 3) + 1 = 4x - 1 have for x in the interval [-10, 10]?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Determine the number of intersection points between the functions f(x) = 3sin(3π(1|((3x - 3)/(-1x + 2)) - 3|)) + 3 and g(x) = (1(-2|x - 3| + 1) - 2)/(2(-2|x - 3| + 1) - 3) where -10 ≤ x ≤ 10.

Present the answer in LaTex format: \boxed{Your answer}


[User]
How many points of intersection exist between the functions f(x) = -3((3(-3x + 4) + 1)/(-2(-3x + 4) - 1))^2 + 5((3(-3x + 4) + 1)/(-2(-3x + 4) - 1)) + 5 and g(x) = -3((-2x - 2)/(1x - 2))^2 + 3((-2x - 2)/(1x - 2)) - 5 in the range -10 ≤ x ≤ 10?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -1|(5((-1(5x^2 + 1x + 2) + 3)/(-2(5x^2 + 1x + 2) + 3)) + 3) + 1| - 2 and g(x) = (2(-3cos(2π(3x^2 - 2x - 3) - 2) + 1) - 3)/(1(-3cos(2π(3x^2 - 2x - 3) - 2) + 1) + 2) intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}


[User]
Find the number of times the graphs of f(x) = -2(-2|(4((1x)/(2x + 3)) - 4) - 3| + 2)^2 - 3(-2|(4((1x)/(2x + 3)) - 4) - 3| + 2) + 1 and g(x) = -1((3(2cos(3πx - 3) + 1) + 1)/(3(2cos(3πx - 3) + 1) - 1))^2 - 4((3(2cos(3πx - 3) + 1) + 1)/(3(2cos(3πx - 3) + 1) - 1)) + 4 intersect in the interval [-10, 10].

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[User]
How many solutions does the equation -3(-2((-3(-1|x - 2| - 3) + 3)/(-2(-1|x - 2| - 3)))^2 - 4((-3(-1|x - 2| - 3) + 3)/(-2(-1|x - 2| - 3)))) + 3 = 2(-1|((1x - 3)/(3x)) - 2|)^2 + 5(-1|((1x - 3)/(3x)) - 2|) + 4 have for x in the interval [-10, 10]?

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[User]
Determine the number of intersection points between the functions f(x) = 3|(2((-1(-2e^(-2x) + 2) - 3)/(2(-2e^(-2x) + 2) + 3))^2 + 2((-1(-2e^(-2x) + 2) - 3)/(2(-2e^(-2x) + 2) + 3)) + 3) + 2| + 1 and g(x) = 1(-5(-1sin(2πx + 1)))^2 + 1(-5(-1sin(2πx + 1))) + 1 where -10 ≤ x ≤ 10.

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[User]
How many points of intersection exist between the functions f(x) = -2|(-3e^(-1((1x + 2)/(1x + 1))) - 1) + 1| and g(x) = -1sin(3π(3x - 1) - 1) in the range -10 ≤ x ≤ 10?

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[User]
How many solutions does the equation 3(1(2|x + 2| - 3) + 1)^2 - 2(1(2|x + 2| - 3) + 1) - 2 = -2sin(2π(2|x + 3| + 1) - 3) - 1 have for x in the interval [-10, 10]?

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[User]
Determine the number of intersection points between the functions f(x) = (-2(2e^(1(5(-4x - 4)^2 - 5(-4x - 4) - 1)) + 1) - 3)/(-1(2e^(1(5(-4x - 4)^2 - 5(-4x - 4) - 1)) + 1) - 2) and g(x) = (-3(-1(5x^2 - 3x + 2) + 4) - 2)/(-2(-1(5x^2 - 3x + 2) + 4) - 3) where -10 ≤ x ≤ 10.

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[User]
Find the number of times the graphs of f(x) = -4((1(-1(-3sin(3πx - 3))^2 + 1(-3sin(3πx - 3)) + 2) + 1)/(-1(-1(-3sin(3πx - 3))^2 + 1(-3sin(3πx - 3)) + 2) - 3)) - 1 and g(x) = (-1(-3(-3sin(1πx - 2) - 3)^2 - 3(-3sin(1πx - 2) - 3) + 3) - 3)/(-2(-3(-3sin(1πx - 2) - 3)^2 - 3(-3sin(1πx - 2) - 3) + 3) - 1) intersect in the interval [-10, 10].

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[User]
Find the number of times the graphs of f(x) = 3(2x - 1)^2 + 4(2x - 1) - 1 and g(x) = (-2x + 2)/(3x - 2) intersect in the interval [-10, 10].

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[User]
Find the number of times the graphs of f(x) = 2|(2x^2 - 1x - 5) + 2| and g(x) = 3|x - 3| - 3 intersect in the interval [-10, 10].

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[User]
How many points of intersection exist between the functions f(x) = -5((3(-3(2|x - 3| - 3) - 1) - 3)/(3(-3(2|x - 3| - 3) - 1)))^2 - 5((3(-3(2|x - 3| - 3) - 1) - 3)/(3(-3(2|x - 3| - 3) - 1))) + 4 and g(x) = (-3(3|(-1x + 1)| + 3) - 2)/(-3(3|(-1x + 1)| + 3) - 2) in the range -10 ≤ x ≤ 10?

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[User]
How many points of intersection exist between the functions f(x) = 2((2(-3cos(1πx + 2) - 3) - 3)/(-3(-3cos(1πx + 2) - 3) + 1))^2 - 3((2(-3cos(1πx + 2) - 3) - 3)/(-3(-3cos(1πx + 2) - 3) + 1)) - 5 and g(x) = (1(-1x + 5) - 3)/(2(-1x + 5) + 1) in the range -10 ≤ x ≤ 10?

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[User]
Find the number of times the graphs of f(x) = (2(4(-1x^2 + 1x + 1) + 5) - 1)/(-1(4(-1x^2 + 1x + 1) + 5)) and g(x) = -5(-2|x + 2| + 3) - 4 intersect in the interval [-10, 10].

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[User]
Determine the number of intersection points between the functions f(x) = -2|(-3((-1(1x^2 + 1) + 1)/(1(1x^2 + 1) - 2)) - 4) - 3| - 1 and g(x) = -2((3(-1|x - 1| - 1) + 1)/(-3(-1|x - 1| - 1) + 2)) - 5 where -10 ≤ x ≤ 10.

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[User]
How many points of intersection exist between the functions f(x) = (3(-3|(-3(1x - 1)^2 + 5(1x - 1) + 3) - 1| - 1) + 1)/(-2(-3|(-3(1x - 1)^2 + 5(1x - 1) + 3) - 1| - 1) - 2) and g(x) = -3|(-2cos(2π(5x - 3) + 1) + 3) - 1| + 1 in the range -10 ≤ x ≤ 10?

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[User]
How many solutions does the equation (1(-2|(-3sin(1πx + 1) + 2)|) + 1)/(1(-2|(-3sin(1πx + 1) + 2)|)) = 1(3cos(2πx + 2) - 2)^2 - 3(3cos(2πx + 2) - 2) + 1 have for x in the interval [-10, 10]?

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[User]
Determine the number of intersection points between the functions f(x) = -4(-3cos(3π(1x^2 + 2x - 3)) - 2) + 4 and g(x) = -2|(3x - 3) - 3| + 1 where -10 ≤ x ≤ 10.

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[User]
How many solutions does the equation 3cos(1π((-2(-4(-3|x - 3| + 1)^2 + 1(-3|x - 3| + 1)) + 1)/(1(-4(-3|x - 3| + 1)^2 + 1(-3|x - 3| + 1)) - 2)) - 2) + 1 = -2cos(3π((-3(-1|x + 3| + 3) + 3)/(-3(-1|x + 3| + 3) + 2)) - 1) - 1 have for x in the interval [-10, 10]?

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[User]
How many points of intersection exist between the functions f(x) = 5(1x^2 + 3x - 5) - 2 and g(x) = -2x^2 + 5x in the range -10 ≤ x ≤ 10?

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[User]
Find the number of times the graphs of f(x) = -2(-1(3|((2x)/(-1x + 3)) + 3| - 3)^2 - 3(3|((2x)/(-1x + 3)) + 3| - 3) + 4) and g(x) = 3|(-1(1x^2 + 5x + 2) + 3) + 3| + 1 intersect in the interval [-10, 10].

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[User]
Determine the number of intersection points between the functions f(x) = -1(5(1|((1x)/(-3x + 1)) + 2| + 1) - 5)^2 + 5(5(1|((1x)/(-3x + 1)) + 2| + 1) - 5) + 2 and g(x) = -2e^(2(5(-3x - 4)^2 - 2(-3x - 4) - 3)) + 3 where -10 ≤ x ≤ 10.

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[User]
Determine the number of intersection points between the functions f(x) = 1|(2(1(-1sin(3πx - 1) - 3)^2 - 5(-1sin(3πx - 1) - 3) - 5) - 5)| - 1 and g(x) = 1|((3(-1x + 4) + 1)/(3(-1x + 4) - 2)) - 3| - 1 where -10 ≤ x ≤ 10.

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[User]
How many solutions does the equation -1sin(2π(-2x^2 - 1x + 1) - 2) - 2 = (2x + 2)/(2x + 3) have for x in the interval [-10, 10]?

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[User]
Find the number of times the graphs of f(x) = -1|((1(3(-4x^2 - 3x - 2) - 5) + 1)/(2(3(-4x^2 - 3x - 2) - 5) - 3)) + 3| and g(x) = -2(1cos(2π(2|x + 1| + 1) + 2)) - 5 intersect in the interval [-10, 10].

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[User]
Find the number of times the graphs of f(x) = -2|(-2((-3x - 1)/(-1x + 3))^2 - 4((-3x - 1)/(-1x + 3)) + 5) + 3| + 1 and g(x) = 2|(-1x - 5) + 2| - 3 intersect in the interval [-10, 10].

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[User]
How many points of intersection exist between the functions f(x) = -1sin(2π(3|x - 1| - 1) + 3) - 1 and g(x) = 1x^2 + 4x in the range -10 ≤ x ≤ 10?

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[User]
Find the number of times the graphs of f(x) = -1|((-2(-2x^2 + 3x + 2) - 2)/(2(-2x^2 + 3x + 2) + 3)) + 1| + 1 and g(x) = -2cos(2π(-3|x + 2|) - 1) - 1 intersect in the interval [-10, 10].

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[User]
Find the number of times the graphs of f(x) = 2|((2(5(3x^2 + 3)) - 3)/(3(5(3x^2 + 3)) + 1)) + 1| and g(x) = (2(1(-2sin(2πx + 2) - 3)^2 - 3(-2sin(2πx + 2) - 3) - 5) - 2)/(-3(1(-2sin(2πx + 2) - 3)^2 - 3(-2sin(2πx + 2) - 3) - 5) - 3) intersect in the interval [-10, 10].

Present the answer in LaTex format: \boxed{Your answer}