Problem 111 - Algebra and Trigonometry Real Mathematics, Real People

Question

Prove that a function of the following form is odd. $$y=a_{2 n+1} x^{2 n+1}+a_{2 n-1} x^{2 n-1}+\cdots+a_{3} x^{3}+a_{1} x$$

Step-by-Step Solution

Answer

The given function \(y=a_{2 n+1} x^{2 n+1}+a_{2 n-1} x^{2 n-1}+\cdots+a_{3} x^{3}+a_{1} x\) is indeed an odd function, because it meets the condition f(-x) = -f(x).

Step 1: Rewrite the Function

Start by writing the general form of the function as \(y=\sum_{n=0}^{N} a_{2 n+1} x^{2 n+1}\), where N is a positive integer and \(a_{2 n + 1}\) is an arbitrary constant and \(x^{2 n + 1}\) is raising \(x\) to an odd power.

Step 2: Substitute -x for x

Substitute \(-x\) for \(x\) in this function, this will give: \( f(-x) = \sum_{n=0}^{N} a_{2 n+1} (-x)^{2 n+1} = \sum_{n=0}^{N} a_{2 n+1} (-1)^{2 n+1}x^{2 n+1}\). This simplifies to \(-\sum_{n=0}^{N} a_{2 n+1}x^{2 n+1}\).

Step 3: Check if f(-x) = -f(x)

As we can see, when we substituted \(-x\) for \(x\), we got the negative of the original function \(-\sum_{n=0}^{N} a_{2 n+1}x^{2 n+1}\), which is exactly \(-f(x)\). Therefore, the given function is an odd function since it meets the condition f(-x) = -f(x).

Key Concepts

Polynomial Function

A polynomial function is an expression made up of terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. These functions are expressed in the general form:

• The form typically looks like this: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\)
• Each term, such as \(a_nx^n\), is a product of a coefficient \(a_n\) and a variable \(x\) raised to a power \(n\).

Polynomial functions can be identified by their degrees, which is determined by the highest power of the variable present in the expression. These are basic building blocks in algebra, often used in various kinds of calculations.
The specific function in the exercise involves odd powers of \(x\), which fundamentally influences its behavior, especially in relation to the function's symmetry.

Function Transformation

Function transformations involve changing the appearance of a function's graph. These transformations can include translations, reflections, stretches, and compressions. When dealing with functions, these transformations can alter its shape, position, and orientation.

• Transformations can describe shifts parallel to the axes, reflections across axes, or stretches.
• The expression \(f(-x)\) typically indicates a reflection over the y-axis.

In our problem, substituting \(-x\) for \(x\) is a type of transformation. This specific transformation helps determine if a function is odd. By reflecting the function across the y-axis, we test its symmetry about the origin, which is a defining trait of odd functions. These manipulations play a crucial role in understanding the characteristics of various classes of functions.

Odd Power Terms

In polynomial functions, odd power terms are expressions where the variable is raised to an odd integer exponent, such as \(x^3\) or \(x^5\). These play a significant role in defining the overall behavior and shape of the function's graph.

• Odd power terms reflect that when \(x\) is replaced by \(-x\), the result is negative, leading to reflection properties.
• The sum of odd powered terms ensures that if the original function \(f(x)\) is given, then \(f(-x)\) will equate to \(-f(x)\).

The nature of these terms allows us to easily analyze if a function is odd by checking if all individual terms meet this condition upon substituting \(-x\). In our exercise, each term in the function notably impacts whether the entire polynomial will be classified as an odd function.

Symmetry in Functions

Symmetry in functions is an essential property used to determine the type of function. A function can be either even, odd, or have no symmetry. In this context, for a function to be considered odd, it must satisfy the condition \(f(-x) = -f(x)\). This indicates a type of symmetry known as rotational symmetry about the origin.

• If the graph of the function looks the same after a 180-degree rotation around the origin, the function is indeed an odd function.
• This symmetry helps in understanding and predicting the behavior of the function across different inputs.

In our specific exercise, determining the oddness of the given polynomial function is a direct application of this symmetry condition. Recognizing and proving such symmetry is crucial in many areas of mathematics, simplifying analysis and solving problems.