Problem 73 - Algebra and Trigonometry

Question

Determine whether the function is even, odd, or neither. \(f(x)=x^{2} \sin x\)

Step-by-Step Solution

Answer

The function is odd.

Step 1: Understand the Definitions

A function is **even** if \(f(-x) = f(x)\) for all \(x\). It is **odd** if \(f(-x) = -f(x)\) for all \(x\). If neither of these conditions holds, the function is neither even nor odd.

Step 2: Calculate f(-x)

Given \(f(x) = x^2 \sin x\), let's find \(f(-x)\):\[ f(-x) = (-x)^2 \sin(-x) = x^2 (-\sin x) = -x^2 \sin x \]

Step 3: Compare f(x) and f(-x)

We have \(f(x) = x^2 \sin x\) and \(f(-x) = -x^2 \sin x\). Comparing these, we check:- \(f(-x) = f(x)\) does not hold since \(-x^2 \sin x eq x^2 \sin x\).- \(f(-x) = -f(x)\) does hold since \(-x^2 \sin x = -(x^2 \sin x)\).

Step 4: Conclusion

Since \(f(-x) = -f(x)\) holds for all \(x\), the function \(f(x) = x^2 \sin x\) is an **odd** function.

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, especially in the realm of calculus and geometry. These functions include sine, cosine, and tangent, along with their reciprocals, cosecant, secant, and cotangent. In the given exercise, the function includes the sine function, denoted as \( \sin x \).

• The sine function \( \sin x \) is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) units.
• It is an odd function, which means that \( \sin(-x) = -\sin x \).
• This property of oddness is crucial in exercises where we determine the symmetry of composite functions that include \( \sin x \).

When combined with other functions, such as the quadratic function \( x^2 \), it influences the overall symmetry of the resulting function. Understanding how trigonometric functions behave on their own helps us predict what happens when they are part of more complex expressions.

Function Symmetry

Function symmetry helps us determine whether a function is even, odd, or neither. This understanding can simplify calculations and graphing.

• An even function has reflective symmetry about the y-axis, and it satisfies the condition that \( f(-x) = f(x) \).


• An odd function has rotational symmetry around the origin. For odd functions, \( f(-x) = -f(x) \) must hold.

In the example exercise, the function \( f(x) = x^2 \sin x \) combines \( x^2 \), an even function, with \( \sin x \), an odd function. While \( x^2 \) would satisfy \( f(-x) = f(x) \), the presence of \( \sin x \) results in \( f(-x) = -f(x) \), making \( f(x) \) an odd function overall.
Understanding function symmetry not only aids in diagnosing function types but also supports graphing techniques and integration practices, as certain symmetrical properties can simplify processes.

Algebraic Functions

Algebraic functions are constructed using polynomials, roots, and ratios of these expressions. These functions frequently appear in algebra and calculus.

• In our exercise, \( x^2 \) represents a simple polynomial, which is an algebraic function. It's utilized in combination with a trigonometric function.


• Algebraic functions like \( x^2 \) have straightforward even properties, evident in \( f(x) \) because \( x^2 \) does not change when \( x \) is replaced with \(-x\).
• The interaction between algebraic and trigonometric functions can adjust or completely change the symmetry property of the function.

Grasping how algebraic functions operate independently and in combination with others offers insight into their behavior within more complex structures. Together, they make up countless functions we analyze in mathematics.