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.
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.
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