Answer:
To find the domain of the function (4x - 3) / (x^3 - 2x^2), you need to consider the values of x for which the function is defined.
In this case, there are two things to watch out for:
- The denominator cannot be equal to zero because division by zero is undefined in mathematics.
- If there are any square roots, cube roots, or other even roots in the expression, you need to ensure that the radicand (the value inside the root) is non-negative, as the square root of a negative number is not a real number.
Let's analyze both of these conditions:
1. The denominator is x^3 - 2x^2. To find when this is equal to zero, we can factor out an x^2:
x^3 - 2x^2 = x^2(x - 2)
Now, we set each factor equal to zero and solve for x:
- x^2 = 0 ---> x = 0
- x - 2 = 0 ---> x = 2
So, the denominator is equal to zero at x = 0 and x = 2. Therefore, the function is undefined at these points.
2. Since there are no square roots, cube roots, or other even roots in the expression, we don't have to worry about the radicand being non-negative.
Now, considering both conditions, the domain of the function is all real numbers except for 0 and 2. In interval notation, the domain can be expressed as:
(-∞, 0) ∪ (0, 2) ∪ (2, ∞)
This means that the function is defined for any real number except 0 and 2.