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 rational expressions, we need to find a common denominator. In this case, the common denominator is (x^2 - 1)(x + 1).
First, let's simplify the individual fractions:
-2 / (x^2 - 1) = -2 / ((x + 1)(x - 1))
x / (x + 1) remains the same.
Now, we can rewrite the expression with the common denominator:
f(x) = (-2 / ((x + 1)(x - 1))) + (x / (x + 1))
To combine the fractions, we need to multiply the numerator of the second fraction by (x - 1) and the numerator of the first fraction by (x + 1):
f(x) = (-2 / ((x + 1)(x - 1))) + ((x(x - 1)) / ((x + 1)(x - 1)))
Now, we can simplify the numerator:
f(x) = (-2 + (x^2 - x)) / ((x + 1)(x - 1))
Combining like terms in the numerator:
f(x) = (x^2 - x - 2) / ((x + 1)(x - 1))
The domain of a rational function is the set of all real numbers excluding the values that make the denominator zero. In this case, we need to avoid (x + 1)(x - 1) = 0.
Setting each factor equal to zero:
x + 1 = 0 --> x = -1
x - 1 = 0 --> x = 1
Therefore, the domain of this function is all real numbers except x = -1 and x = 1. We exclude these values because they would make the denominator zero, resulting in undefined values.