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Chapter 1: Problem 74

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$h(x)=x \sqrt{x+5}$$

Short Answer

Expert verified
The function \( h(x) = x \sqrt{x+5} \) is neither even nor odd, and does not possess any kind of symmetry.

Step by step solution

01

Identify Function Type

First, it's necessary to replace \( x \) with \( -x \) in the function \( h(x) = x \sqrt{x+5} \), to conclusively establish the type of function being dealt with. This gives \( h(-x) = -x \sqrt{-x+5} \).
02

Comparisons of Original and Transformed Functions

The next task involves checking if \( h(x) = h(-x) \) or \( h(x) = -h(-x) \). However, it's evident that \( x \sqrt{x+5} \) is not equal to \( -x \sqrt{-x+5} \), so the function isn't even. In addition, \( x \sqrt{x+5} \) doesn't match \( -(-x \sqrt{-x+5}) \), meaning that the function isn't odd as well.
03

Final Determination of Symmetry

Given that the function \( h(x) = x \sqrt{x+5} \) fails to be categorised as either odd or even, we can definitively say that it possesses neither symmetry about the y-axis nor origin. This leads to the conclusion that \( h(x) \) is neither an odd nor even function and does not depict a symmetry.

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