Find the Domain f(x) = square root of x^2-1

Problem

ƒ(x)=√(,x2−1)

Solution

1. Identify the restriction for a square root function, which requires the radicand to be greater than or equal to zero.

x2−1≥0

2. Factor the quadratic expression using the difference of squares formula.

(x−1)*(x+1)≥0

3. Determine the critical points by setting each factor to zero, which gives x=1 and x=−1

x−1=0⇒x=1

x+1=0⇒x=−1

4. Test the intervals created by the critical points (−∞,−1] [−1,1] and [1,∞) to see where the inequality holds true.

For *x=−2:(−2)2−1=3≥0

For *x=0:(0)2−1=−1<0

For *x=2:(2)2−1=3≥0

5. Conclude that the inequality is satisfied when x≤−1 or x≥1

x∈(−∞,−1]∪[1,∞)

Final Answer

Domain: *(−∞,−1]∪[1,∞)

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