Determine if Odd, Even, or Neither f(x)=x square root of x+5

Problem

ƒ(x)=x√(,x+5)

Solution

1. Identify the domain of the function. For the square root to be defined, the radicand must be non-negative: x+5≥0 which means x≥−5

2. Check for symmetry in the domain. A function can only be even or odd if its domain is symmetric about the origin (i.e., if x is in the domain, then −x must also be in the domain).

3. Evaluate the domain at a specific point. For example, x=5 is in the domain because 5≥−5 However, x=−6 is not in the domain because −6<−5

4. Conclude based on the domain. Since the domain [−5,∞) is not symmetric about the origin, the condition ƒ*(−x)=ƒ(x) or ƒ*(−x)=−ƒ(x) cannot hold for all x in the domain.

Final Answer

ƒ(x)=x√(,x+5)* is Neither

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