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 find the domain of a function, we need to determine the set of all possible input values (or x-values) for which the function is defined. In this case, we need to consider two things:

1. The denominator, which cannot be equal to zero, as division by zero is undefined. Therefore, we need to find the values of x for which the denominator, 15x^2 + 2x - 8, is not equal to zero.

2. The radicand (the expression inside the square root), which must be greater than or equal to zero, as the square root of a negative number is not defined in the real number system.

Let's solve these conditions step by step:

1. For the denominator to be non-zero, we can set it equal to zero and solve for x:
15x^2 + 2x - 8 ≠ 0

To solve this quadratic inequality, you can use factoring, completing the square, or the quadratic formula. Let's solve it by factoring:

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

Setting each factor equal to zero, we have:
3x - 2 = 0 or 5x + 4 = 0

Solving for x in each equation gives us:
3x = 2 or 5x = -4
x = 2/3 or x = -4/5

Therefore, x cannot be equal to 2/3 or -4/5, as that would make the denominator zero. So, the domain of the function excludes these values.

2. Next, let's consider the radicand, 9x^2 - 12x + 4. We need this expression to be greater than or equal to zero:

9x^2 - 12x + 4 ≥ 0

To solve this quadratic inequality, we can use factoring, completing the square, or the quadratic formula. Let's solve it by factoring:

9x^2 - 12x + 4 = 0
(3x - 2)(3x - 2) = 0

Setting the factor equal to zero, we have:
3x - 2 = 0

Solving for x gives us:
3x = 2
x = 2/3

Since this quadratic expression is always zero or positive, it means that for any value of x, the radicand is greater than or equal to zero. Therefore, there are no restrictions on x due to the radicand.

Combining the two sets of restrictions, we find that the domain of the function f(x) is all real numbers except x = 2/3 and x = -4/5.

To find the absolute value of this domain, we simply take the positive values of the excluded x-values. Therefore, the absolute value of the domain is |x| ≠ 2/3 and |x| ≠ 4/5.