Sets on builder notations

Question

f(x) = 1√|x|²−|x|−6

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Find the domain f(x) =1/√|x|²−|x|−6 of write the answers in builder notation.

Answer 1

\(f(x) = \frac1{\sqrt{|x|^2 - |x|-6}}\)

For domain \(|x|^2 - |x | - 6 > 0\) 

⇒ \((|x |-3) (|x| + 2) > 0\)

⇒ \(|x| < -2 \;or \; |x |>3\)

(not possible)

⇒ \(|x| > 3\)

⇒ \(x \in (-\infty,-3)\cup (3, \infty)\)

Domain = \( (-\infty,-3)\cup (3, \infty)\)

\(= \{x |x\in R \text{ & } |x| > 3\}\)