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An even function is a type of function that exhibits reflectional symmetry about the y-axis. This means that the graph of the function remains unchanged if you flip it over the y-axis.

Simply put, if a function is even, the following condition holds true:

\[ f(-x) = f(x) \]What this tells us is that substituting \(-x\) into the function gives the same result as substituting \(x\).

Here's an easy way to visualize it:

• The graph looks the same to the left and right of the y-axis.
• Classic examples include functions like \(f(x) = x^2\).

In the exercise provided, when we replaced \(x\) with \(-x\), the outcome \(-x^3 + x\) did not match the original function \(x^3 - x\). Therefore, the function \(f(x) = x^3 - x\) is not an even function, as it does not satisfy the condition \(f(-x) = f(x)\).

A polynomial function is a mathematical expression involving a sum of powers in one or more variables, each multiplied by a coefficient.

It typically has the form:\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \]Here \(a_n, a_{n-1}, \ldots, a_0\) are constants, and \(n\) is a non-negative integer which represents the highest degree of the polynomial.

Now let's connect this idea to the specific problem: the function \(f(x) = x^3 - x\) is a polynomial function of degree 3 because the highest power of \(x\) is 3. A polynomial of odd degree might have interesting symmetry properties, which brings us to the concept of odd and even functions.

Generally, polynomial functions with even degrees tend to be even functions if there are no missing terms, while those with odd degrees might often be odd functions.

Function symmetry is a way to understand how the graph of a function behaves in relation to the axes.

There are two main types of symmetry we often discuss:

• Even Symmetry (y-axis symmetry): This occurs when the function looks the same on both sides of the y-axis, as we mentioned in the first section.
• Odd Symmetry (origin symmetry): A function is said to have odd symmetry if rotating it 180 degrees around the origin doesn’t change the graph.

To determine odd symmetry, the condition
\(-f(x) = f(-x)\) must hold true.

In our exercise, for \(f(x) = x^3 - x\), we found that \(-f(x)\) is exactly the same as \(f(-x)\) thus confirming its odd symmetry. Understanding symmetry helps in graph sketching, functional analysis, and simplification of complex mathematical problems. Identifying whether a function is even, odd, or neither provides insight into its properties and behavior.