Find the Domain square root of 3-x+ square root of x^2-1

Problem

√(,3−x)+√(,x2−1)

Solution

1. Identify the conditions for the domain of a square root function. For the expression to be defined in the set of real numbers, the radicand (the expression inside the square root) must be greater than or equal to zero.

2. Set up the first inequality for the term √(,3−x)

3−x≥0

3. Solve the first inequality for x

−x≥−3

x≤3

4. Set up the second inequality for the term √(,x2−1)

x2−1≥0

5. Factor the quadratic expression to find the critical points.

(x−1)*(x+1)≥0

6. Determine the intervals where the quadratic inequality is true. The product is non-negative when x is outside the roots −1 and 1

x≤−1* or *x≥1

7. Intersect the solutions from both inequalities to find the common domain. We need x≤3 AND (x≤−1 or x≥1.

(−∞,−1]∪[1,3]

Final Answer

Domain:(−∞,−1]∪[1,3]

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