Problem 41 - Precalculus : Building Concepts and Connections

Question

Decide if each function is odd, even, or neither by using the definitions. $$f(x)=x^{5}-2 x$$

Step-by-Step Solution

Answer

The given function \(f(x)=x^{5}-2x\) is an odd function.

Step 1: Compute f(-x)

To find whether the function is even or odd, we need to calculate \(f(-x)\) which involves replacing every \(x\) in \(f(x)\) with \(-x\). So, \(f(-x)=(-x)^{5}-2(-x)= -x^{5}+2x\).

Step 2: Compare f(-x) with f(x) and -f(x)

Now, compare \(f(-x)\) with \(f(x)\) and \(-f(x)\). We have \(f(x) = x^{5}-2x\) and \(-f(x) = -x^{5}+2x\). Comparing, we can see that \(f(-x)= -f(x)\), hence the given function is an odd function. And it is clear that \(f(-x) \neq f(x)\), so it is not an even function.

Key Concepts

Function Symmetry

Understanding function symmetry is essential in determining if a function is odd, even, or neither. The symmetry of a function reveals how it behaves when its input is transformed (like flipping across axes). A function is said to be **even** if it is symmetric with respect to the y-axis. Mathematically, this means that for every input \(x\), \(f(x) = f(-x)\). Think of it as looking into a mirror placed on the y-axis: the reflection of the graph on one side should perfectly overlap with the other side.

An **odd** function, on the other hand, is symmetric about the origin. This implies that for every \(x\), \(f(-x) = -f(x)\). A useful way to visualize this is by performing a 180-degree rotation around the origin. If the function maps onto itself in this way, it is odd.

If neither symmetry condition is satisfied, the function is neither even nor odd. Understanding these properties allows you to discern more information about the function's behavior and how it might look when graphed.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents, such as \(f(x) = x^5 - 2x\). These functions are quite straightforward since they are composed of several terms added together, each term being a product of a constant and a variable raised to a power.

Polynomials can be classified based on their degree, which is the highest power of the variable in the expression. For example, in \(x^5 - 2x\), the degree is 5, indicating a quintic polynomial. The structure of polynomial functions usually provides visual symmetry clues. **Even-degree polynomials** tend to show even symmetry, while **odd-degree polynomials** often display odd symmetry.

One significant property of polynomials is how they behave with respect to function symmetry. For our original example, calculating \(f(-x)\) yields the expression \(-x^5 + 2x\), leading to a determination that \(f(x)\) is an odd function since \(f(-x)\) mirrors \(-f(x)\). Understanding these fundamental aspects is crucial for correctly identifying the symmetry properties of polynomial functions.

Function Transformation

Function transformation involves modifying a function's graph in a variety of ways. These transformations can include shifting, scaling, reflecting, or rotating. When analyzing a function like \(f(x) = x^5 - 2x\), transformations can provide insights into behavior changes without altering the function’s inherent properties.

Here are common types of transformations:

• **Translation**: Shifting the graph horizontally or vertically depending on added constants outside the function.
• **Scaling**: Expanding or contracting the graph in the vertical or horizontal direction, affecting the steepness or width of the graph shape.
• **Reflection**: Flipping the graph over an axis. This forms the core of determining whether a function is even or odd.

Reflecting a function about the y-axis or the origin can help reveal its symmetry characteristics, aiding the determination of evenness or oddness. Through transformations, the essential nature of functions becomes more visually and conceptually clear, offering deeper insights into their mathematical properties.