Final answer:
The coordinates of the hole in the function f(x) = (3x^2 - 11x + 10)/(4x^2 - 7x - 2) are (-1/4, -1) and (2, 2).
Step-by-step explanation:
To find the coordinates of the hole, we need to determine the values of x for which the denominator of the function becomes zero. In this case, the denominator is 4x^2 - 7x - 2. Setting it equal to zero and solving for x, we get:
4x^2 - 7x - 2 = 0
Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), where a = 4, b = -7, and c = -2. Plugging in the values:
x = (-(-7) ± √((-7)^2 - 4(4)(-2)))/(2(4))
x = (7 ± √(49 + 32))/(8)
x = (7 ± √81)/(8)
x = (7 ± 9)/(8)
Therefore, the possible values for x are: x = (-2/8) or x = (16/8)
x = -1/4 or x = 2
Adding these values back into the original equation to find the corresponding y-coordinates:
For x = -1/4, y = (3(-1/4)^2 - 11(-1/4) + 10)/(4(-1/4)^2 - 7(-1/4) - 2) = -1
For x = 2, y = (3(2)^2 - 11(2) + 10)/(4(2)^2 - 7(2) - 2) = 2
Therefore, the coordinates of the hole are (-1/4, -1) and (2, 2).