Problem 19 - Precalculus : Building Concepts and Connections

Question

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$f(x)=-x^{3}+2 x$$

Step-by-Step Solution

Answer

The given function \(f(x) = -x^{3} + 2x\) is neither an odd nor an even function.

Step 1: Identify the Function

The given function is \(f(x) = -x^{3} + 2x\). This is a polynomial of degree 3, where each term has its own variable \(x\).

Step 2: Check for Even Symmetry

Replace \(x\) with \(-x\) in the function, and simplify. The resulting function is \(f(-x) = -(-x)^{3} + 2(-x) = -x^{3} - 2x\). This is not equal to the original function, so the function is not even.

Step 3: Check for Odd Symmetry

If a function is not even, it could be odd. A function is odd if \(f(-x)\) is equal to \(-f(x)\). In this case, \(-f(x) = -(-x^{3} + 2x) = x^{3} - 2x\), which is not equal to \(f(-x)\). Therefore, the function is not odd either.

Key Concepts

Even Functions

Even functions have a special kind of symmetry, making them visually intuitive once you understand the concept. Their graph is symmetric concerning the y-axis. This means if you fold the graph along the y-axis, both sides will match perfectly. To determine if a function is even, we examine if the function satisfies the equation:

• \( f(x) = f(-x) \)

This means replacing every instance of \( x \) with \( -x \), and seeing if the new function is identical to the original.
For example, consider \( f(x) = x^2 \). By substituting \( -x \) into the function, we get \( f(-x) = (-x)^2 \), which simplifies to \( x^2 \). Thus, \( f(x) = f(-x) \), confirming it's even.
Even functions can often be identified by looking at the powers of their terms. An even function generally contains only even powers of \( x \), although exceptions exist. Understanding even functions is crucial for grasping their symmetrical properties.

Odd Functions

Odd functions bring their unique charm with symmetrical properties around the origin. The graph of an odd function when rotated 180 degrees around the origin will look the same. A function is characterized as odd if it satisfies the equation:

• \( f(-x) = -f(x) \)

Simply, you replace \( x \) with \( -x \) in the function and check if it equals the negative of the original function.
Take, for instance, \( f(x) = x^3 \). Here, substituting \( -x \) results in \( f(-x) = (-x)^3 = -x^3 \), which is \( -f(x) \). Hence, \( x^3 \) is an odd function.
Characteristics of an odd function include having terms where only odd powers of \( x \) are present. This property makes them straightforward to identify and essential for analyzing symmetries.

Polynomial Functions

Polynomial functions are a cornerstone of algebra and calculus, comprised of variables raised to whole number powers and constants. They are expressed in the general form:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

where the coefficients \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( n \) is a non-negative integer.
In classifying polynomial functions, their degree (the highest power of \( x \)) plays a crucial role in determining their behavior and classification as even, odd, or neither.
An example of a polynomial function is \( f(x) = -x^3 + 2x \), where the highest degree is 3. By analyzing its symmetry properties through replacing \( x \) with \( -x \), we can determine that \( f(x) = -x^3 + 2x \) is neither even nor odd, due to its inconsistency with both even and odd symmetry rules.
Understanding polynomials and their symmetries allows for a deeper comprehension of how these functions behave and interact graphically.