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Find the domain of the function $f(x) = \frac{x^2 + 10x + 21}{x^2 - 4x - 21} + \frac{x^2 - 1}{x^2 - 4x + 4}$.  (Express your answer using interval notation.)

 Aug 20, 2023
edited by tomtom  Aug 20, 2023
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This problem involves the addition of rational functions. Since fractions are involved, we have to consider when the denominator equals 0. This process is made easier by the process of factoring.

 

\(f(x) = \frac{x^2 + 10x + 21}{x^2 - 4x -21} + \frac{x^2 - 1}{x^2 - 4x + 4} \\ f(x) = \frac{x^2 + 10x + 21}{(x + 3)(x - 7)} + \frac{x^2 - 1}{(x - 2)^2}\)

 

After factoring the denominators, we can identify the x-values that will be outside of the domain.

 

\(x + 3 \neq 0 \quad x - 7 \neq 0 x - 2 \neq 0 \\ x \neq -3 \quad x\neq 7 \quad x \neq 2\)

 

All of these x-values are not within the domain.

 

Therefore, the domain in interval notation is \((-\infty, -3) \cup (-3, 2) \cup (2, 7) \cup (7, \infty)\)

 Aug 21, 2023

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