Question

Determine whether the functions are even, odd, or neither. a. \(y=\cot x\) b. \(y=2 \sin \frac{x}{2}\) c. \(y=2 \sec x\)

Short answer

\(y=\cot x\) is odd, \(y=2 \sin \frac{x}{2}\) is odd, and \(y=2 \sec x\) is even.

Step-by-step solution

Understand the Function Basics

For each function, replace \(x\) by \(-x\), and see if the resulting function is equivalent to the original function (which would make the function even), equivalent to the negative of the original function (which would make the function odd), or neither.

Determine If \(y = \cot{x}\) Is Even, Odd, Or Neither

Replace \(x\) with \(-x\) in the cotangent function: \(\cot{(-x)} = -\cot{x}\). As this is equivalent to the negative of the original function, \(\cot{x}\) is an odd function.

Determine If \(y = 2 \sin{\frac{x}{2}}\) Is Even, Odd, Or Neither

Replace \(x\) with \(-x\) in the sine function: \(2 \sin{-\frac{x}{2}} = -2 \sin{\frac{x}{2}}\). As this is equivalent to the negative of the original function, \(2 \sin{\frac{x}{2}}\) is also an odd function.

Determine If \(y = 2 \sec{x}\) Is Even, Odd, Or Neither

Replace \(x\) with \(-x\) in the secant function: \(2 \sec{(-x)} = 2\sec{x}\). As this is the same as the original function, \(2 \sec{x}\) is an even function.

Final Answer

So, function \(y = \cot{x}\) is odd, function \(y = 2 \sin{\frac{x}{2}}\) is odd and function \(y = 2 \sec{x}\) is even.