Find the Domain and Range f(x) = square root of x^2-16

Problem

ƒ(x)=√(,x2−16)

Solution

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

x2−16≥0

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

(x−4)*(x+4)≥0

3. Solve the inequality by testing intervals around the roots x=−4 and x=4

x≤−4* or *x≥4

4. Express the domain in interval notation.

Domain: *(−∞,−4]∪[4,∞)

5. Determine the range by observing that the principal square root function √(,u) always produces values greater than or equal to zero.

ƒ(x)≥0

6. Verify the minimum value by substituting a boundary point from the domain, such as x=4

ƒ(4)=√(,4−16)=0

7. Express the range in interval notation.

Range: *[0,∞)

Final Answer

Domain: *(−∞,−4]∪[4,∞), Range: *[0,∞)

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