f(x) = (-2 / (x^2) - 1) + (x / (x + 1))
Add the rationals and then simplify as much as possible. Show each step.
What is the domain of this function? Explain how you know.
To add the rationals, we need a common denominator. Since the denominators are x^2 and x + 1, the common denominator is (x^2)(x + 1).
Rewriting each rational with the common denominator:
f(x) = (-2(x + 1) / (x^2)(x + 1)) + (x(x^2) / (x + 1)(x^2))
Now, we can combine the two fractions:
f(x) = (-2(x + 1) + x(x^2)) / (x^2)(x + 1)
Expanding and simplifying:
f(x) = (-2x - 2 + x^3) / (x^2)(x + 1)
The domain of a rational function is the set of all real numbers except for the values that make the denominator equal to zero, as division by zero is undefined. In this case, the denominator (x^2)(x + 1) can never be equal to zero because the product of non-zero factors is non-zero. Hence, the domain of this function is all real numbers, which can be represented as (-∞, +∞).