Question

Determine whether the function is even, odd, or neither. Use a graphing utility to verify your result. $$ f(x)=x \cos x $$

Short answer

The function \(f(x)=x \cos x\) is an odd function.

Step-by-step solution

Determine if the Function is Even

Apply the condition of an even function \(f(-x) = f(x)\) to the function \(f(x)=x \cos x\). Thus, \(f(-x) = -x \cos(-x) = -x \cos x \neq x \cos x = f(x)\). Therefore, \(f(x)\) is not an even function.

Determine if the Function is Odd

Apply the condition of an odd function \(f(-x) = -f(x)\) to the given function. Thus, \(f(-x) = -x \cos x = -f(x)\). Thus, based on the odd function condition, \(f(x) = x \cos x\) is an odd function.

Verification using a graphing Utility

To confirm that the function is odd, one can graph it using any graphing tool or calculator. For an odd function, its graph has origin symmetry; i.e., it's reflected about the origin. Graphing the function \(f(x) = x \cos x\), you should notice this symmetry, confirming that \(f(x)\) is an odd function.