Final answer:
The domain of the function f(x) = 2x / (2x^3 + 5x^2 - 12x) is all real numbers except for 0, 3/2, and -4, as these are the values that make the denominator zero. So, the correct answer is that the domain is all real numbers such that x is not equal to 0, 3/2, and -4.
Step-by-step explanation:
To determine the domain of the rational function f(x) = 2x / (2x^3 + 5x^2 - 12x), we must find all the values of x for which the function is defined. This means we need to find the x values that do not make the denominator equal to zero because division by zero is undefined.
First, factor the denominator:
- 2x^3 + 5x^2 - 12x = x(2x^2 + 5x - 12)
- Now, factor the quadratic: 2x^2 + 5x - 12
- This factors to (2x-3)(x+4)
Now, set each factor that contains an x to zero and solve:
- x = 0
- 2x - 3 = 0 → x = 3/2
- x + 4 = 0 → x = -4
Therefore, the domain of f(x) is all real numbers except for the values that make the denominator zero, which are 0, 3/2, and -4.
The correct answer to the domain of the function f(x) = 2x / (2x^3 + 5x^2 - 12x) is:
- x is an element of all real numbers such that x is not equal to 0, 3/2, and -4.
Therefore, option 4 is correct.