Find the Domain f(x)=( square root of x-2)/(x^2-10)

Problem

ƒ(x)=√(,x−2)/(x2−10)

Solution

1. Identify the restriction for the square root. For the expression √(,x−2) to be defined in the real numbers, the radicand must be non-negative.

x−2≥0

2. Solve the inequality for the numerator.

x≥2

3. Identify the restriction for the denominator. The denominator of a fraction cannot be equal to zero.

x2−10≠0

4. Solve for the values that make the denominator zero.

x2=10

x=±√(,10)

5. Determine the intersection of the conditions. We must satisfy x≥2 and x≠±√(,10) Since −√(,10) is less than 2 it is already excluded by the first condition. However, √(,10)≈3.16 is greater than 2 so it must be specifically excluded from the interval.

x∈[2,√(,10))∪(√(,10),∞)

Final Answer

Domain:[2,√(,10))∪(√(,10),∞)

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