Problem 64 - Precalculus: A Concise Course

Question

Use the graph of the function to determine whether the function is even, odd, or neither. Verify your answer algebraically. $$ g(x)=x^{2} \cot x $$

Step-by-Step Solution

Answer

The function \(g(x) = x^2 \cot(x)\) is neither even nor odd.

Step 1: Checking if the function is even

Replace \(x\) with \(-x\) in the equation \(g(x) = x^2 \cot(x)\) and simplify. This gives \(g(-x) = (-x)^2 \cot(-x) = x^2 (-\cot(x)) = -x^2 \cot(x)\). Since \(g(-x) \neq g(x)\), \(g(x)\) is not an even function.

Step 2: Checking if the function is odd

The definition of an odd function is \(f(x) = -f(-x)\). However, we have already found that \(g(-x) = -x^2 \cot(x)\), which is not equal to \(-g(x) = -x^2 \cot(x)\). Therefore, \(g(x)\) is not an odd function.

Step 3: Conclusion

Since \(g(x)\) is neither even nor odd, the function is neither.

Key Concepts

Function Symmetry

Functions can be symmetrical in different ways, leading to interesting properties and insights into their behavior. Symmetry is all about the geometric relationship of the graph of the function between the positive and negative sides of the x-axis.
Even functions exhibit symmetry about the y-axis. This means if you took one side of the graph and flipped it over the y-axis, it would match perfectly with the other side. Mathematically, even functions satisfy the property:

• \( f(-x) = f(x) \) for all \( x \) in the domain.

Odd functions, on the other hand, have symmetry about the origin. This basically transforms the entire graph through the origin. To be precise, if you rotate the graph 180 degrees around the origin, it should look the same. Odd functions meet the following criteria:

• \( f(-x) = -f(x) \) for all \( x \) in their domain.

With this understanding of symmetry, identifying whether a function is even, odd, or neither becomes a systematic task.

Algebraic Verification

Algebraic verification is a powerful approach that involves using algebraic methods to confirm the symmetry of functions. This is done by substituting \( -x \) for \( x \) in the function's expression and simplifying.
This process is simple but effective:

• For even functions, after replacing \( x \) with \( -x \), the equation should revert back to its original form: \( f(-x) = f(x) \).
• For odd functions, you should end up with the negative of the original expression: \( f(-x) = -f(x) \).

When neither of these conditions is satisfied, the function is neither even nor odd, as with the example function \( g(x) = x^2 \cot(x) \). By substituting \( -x \) into this function, we get \( g(-x) = (-x)^2 \cot(-x) = x^2 (-\cot(x)) = -x^2 \cot(x) \), proving neither condition aligns with its respective symmetry requirement.

Trigonometric Functions

Trigonometric functions are more than just sines and cosines; they include a range of functions like tangent, cotangent, secant, and cosecant. Understanding their properties is essential for determining function symmetry.
Let's take a closer look at the cotangent function \( \cot(x) \), which is the focus of our original problem. The cotangent function is defined as the reciprocal of the tangent function:

• \( \cot(x) = \frac{1}{\tan(x)} \)
• It is undefined at integer multiples of \( \pi \), where the sine component becomes zero.

When dealing with even and odd function verification, note that \( \cot(-x) = -\cot(x) \), indicating the odd nature of the cotangent itself. However, when combined with other components, such as \( x^2 \) in the problem \( g(x) = x^2 \cot(x) \), the whole function is affected. Though \( \cot(x) \) is odd, \( x^2 \) is even, resulting in a combination that doesn't meet the full criteria for either symmetry alone.
This convergence of trigonometric properties and algebraic manipulation underscores the complexity and beauty of analyzing function symmetry.