Problem 73 - Precalculus Mathematics for Calculus

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: Define Even Function

A function is called even if for every \(x\) in its domain, \(f(-x) = f(x)\). Mathematically, an even function is symmetric with respect to the y-axis.

Step 2: Define Odd Function

A function is called odd if for every \(x\) in its domain, \(f(-x) = -f(x)\). Mathematically, an odd function is symmetric with respect to the origin.

Step 3: Calculate \(f(-x)\)

Given \(f(x) = x^2 \sin x\), we need to calculate \(f(-x)\). So, replace every \(x\) with \(-x\): \[ f(-x) = (-x)^2 \sin(-x) \]This simplifies to:\[ f(-x) = x^2 (-\sin x) \]So, \(f(-x) = -x^2 \sin x\).

Step 4: Check for Even Function

To determine if the function is even, we compare \(f(-x)\) with \(f(x)\): \[ f(-x) = -x^2 \sin x \quad \text{and} \quad f(x) = x^2 \sin x \]Since \(f(-x) eq f(x)\), the function is not even.

Step 5: Check for Odd Function

To determine if the function is odd, we compare \(f(-x)\) with \(-f(x)\): \[ f(-x) = -x^2 \sin x \quad \text{and} \quad -f(x) = -x^2 \sin x \]Since \(f(-x) = -f(x)\), the function is odd.

Key Concepts

Symmetry of Functions

The symmetry of a function is a key concept in precalculus mathematics. It helps us understand how a function behaves in relation to certain axes or points. There are two primary types of symmetry to consider: symmetry with respect to the y-axis and symmetry with respect to the origin.

- **Symmetry with respect to the y-axis:** A function is symmetric with respect to the y-axis if for any point \ \(x\) on the graph, the point \(-x\) gives the same function value \(f(x)\). This means that the left side of the y-axis mirrors the right side.
- **Symmetry with respect to the origin:** A function is symmetric with respect to the origin if rotating the graph 180 degrees around the origin results in the same graph. For this to happen, if a point \((x, y)\) exists on the graph, then \((-x, -y)\) must also exist.

Testing for these symmetries is essential in determining if a function is even or odd. An even function shows y-axis symmetry, whereas an odd function exhibits origin symmetry.

Function Properties

Understanding a function's properties can make solving mathematical problems easier. For this exercise, we focus on even and odd functions.

- **Even Functions:** An even function satisfies the condition \(f(-x) = f(x)\) for all \(x\) in the function's domain. Graphically, this results in a mirror-like symmetry along the y-axis. Classic examples include \(f(x) = x^2\) or \(f(x) = \cos(x)\).
- **Odd Functions:** An odd function satisfies the condition \(f(-x) = -f(x)\) for all \(x\). Odd functions are symmetric about the origin, meaning you could rotate the graph 180 degrees and it's unchanged. Examples are \(f(x) = x^3\) or \(f(x) = \sin(x)\).

By understanding these characteristics, you can determine whether a function is even, odd, or neither, aiding in grasping the function's behavior and graph.

Precalculus Mathematics

Precalculus mathematics forms the foundation for studying calculus and involves various fundamental concepts, including function properties like evenness and oddness.

In precalculus:

• Functions are assumed to have certain behaviors, such as continuous graphs and predictable symmetry.
• Identifying symmetries helps solve equations more efficiently, providing insights into the nature of function graphs.
• Understanding these properties ensures a smoother transition into calculus, where the analysis of graphs becomes even more complex.

By focusing on these foundational aspects, students gain the necessary skills to tackle more advanced mathematical problems. Precalculus strengthens not only procedural skills but also critical thinking, an asset in any field that requires mathematical reasoning.