If f(x)= sqrt(9x^2-12x+4)/(15x^2+2x-8),what is d domain of the function. And what is it in absolute value(abs)

To determine the domain of the function f(x), we need to identify any values of x that would result in undefined expressions.

Let's start by analyzing the numerator: sqrt(9x^2 - 12x + 4). The square root function is defined only for non-negative values. Therefore, we need to ensure that the expression inside the square root is non-negative:

9x^2 - 12x + 4 ≥ 0

To solve this inequality, we can find the zeros of the quadratic expression:

9x^2 - 12x + 4 = 0

Using the quadratic formula, x = (-b ± sqrt(b^2 - 4ac))/(2a), where a = 9, b = -12, and c = 4, we can calculate the values of x:

x = [12 ± sqrt((-12)^2 - 4 * 9 * 4)] / (2 * 9)
x = (12 ± sqrt(144 - 144)) / 18
x = (12 ± sqrt(0)) / 18
x = 12 / 18
x = 2/3

Since the square root expression is non-negative (0), the numerator is non-negative for all values of x.

Now, we can look at the denominator: 15x^2 + 2x - 8. We need to ensure that this expression is not equal to zero because division by zero is undefined. We can find the zeros by factoring the quadratic expression:

15x^2 + 2x - 8 = 0
(3x - 2)(5x + 4) = 0

Setting each factor equal to zero, we find:

3x - 2 = 0 --> x = 2/3
5x + 4 = 0 --> x = -4/5

So, the values of x that make the denominator equal to zero are x = 2/3 and x = -4/5.

Now, considering both the numerator and the denominator, we can determine the domain of the function f(x). The domain consists of all values of x, except those that make the denominator zero. Therefore, the domain of f(x) is:

Domain of f(x) = {x | x ≠ 2/3, -4/5}

Now, let's calculate the domain of f(x) in absolute value (abs):

Taking the absolute value of the domain values means we remove the negative sign if it exists. Therefore, the absolute value of the domain of f(x) is:

|Domain of f(x)| = {x | x ≥ 2/3, 4/5}

Please note that in absolute value, the inequality sign switches from "≠" to "≥" to account for the removal of the negative values.