Problem 17 - College Algebra

Question

Determine whether each function is even, odd, or neither. $$f(x)=x^{3}+x$$

Step-by-Step Solution

Answer

The function \(f(x) = x^{3} + x\) is odd.

Step 1: Substitute -x into the equation

Replace \(x\) in the equation \(f(x) = x^{3} + x\) with \(-x\) to get: \(f(-x) = (-x)^{3} - x\).

Step 2: Simplify the equation

Simplify \(f(-x) = (-x)^{3} - x\) to \(f(-x) = -x^3 - x\)

Step 3: Compare with the original function

Now, compare \(f(-x) = -x^3 - x\) with \(f(x) = x^3 + x\) and \(-f(x) = -x^3 - x\). If \(f(-x) = f(x)\), the function is even. If \(f(-x) = -f(x)\), the function is odd.

Step 4: Conclude the type of the function

Given that \(f(-x) = -f(x)\), the function is odd.

Key Concepts

Even and Odd Functions

When it comes to understanding functions, an important aspect is distinguishing between even, odd, and neither types. The even or odd nature of a function tells us about specific symmetries it may possess.

• Even Functions: A function is considered even if, when you replace every occurrence of \(x\) with \(-x\), the function remains unchanged. Mathematically, this means that \(f(-x) = f(x)\) for all \(x\) in the domain of the function. Even functions are symmetric about the y-axis. A visual example would be the parabola described by the function \(f(x) = x^2\).
• Odd Functions: On the contrary, a function is odd if when you substitute \(-x\) for \(x\), the function's expression becomes the opposite sign of the original. That is, if \(f(-x) = -f(x)\), then the function is odd. Odd functions display symmetry about the origin, meaning that if you rotate the graph 180 degrees about the origin, it looks the same. For instance, the cubic function \(f(x) = x^3\) is odd.

In our example with \(f(x) = x^{3} + x\), calculating the function at \(-x\) gives \(f(-x) = -x^3 - x\), which equals \(-f(x)\). Therefore, this function is odd.

Function Symmetry

Symmetry plays a vital role in understanding the behavior of functions. When a function has symmetry, it becomes much easier to predict how changes in one part of the function will affect other parts. Symmetry can generally be categorized into a few types:

• Y-axis Symmetry: This occurs in even functions, where \(f(-x) = f(x)\). The graph of the function mirrors itself across the y-axis. It means the left and right halves of the graph are mirror images.
• Origin Symmetry: This kind of symmetry is characteristic of odd functions where \(f(-x) = -f(x)\). When a graph is symmetric with respect to the origin, rotating it 180 degrees about the origin point keeps the graph the same.

The function \(f(x) = x^3 + x\) displays origin symmetry because \(f(-x) = -x^3 - x = -f(x)\). This trait is typical for odd functions.

Polynomial Functions

Polynomial functions are an essential class of functions that are defined by expressions that involve sums and differences of powers of \(x\). These functions have the general form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where each \(a_i\) represents a coefficient, and \(n\) is a non-negative integer representing the degree of the polynomial.Key features of polynomial functions include:

• Degrees and Terms: The degree is the highest power of \(x\) that appears in the polynomial. Each separate power of \(x\) (like \(x^3\), \(x^2\), etc.) is called a term of the polynomial.
• Behavior: The degree of a polynomial function greatly impacts its graph's shape and behavior. Higher-degree polynomials may have more complex oscillations compared to linear or quadratic polynomials.
• Symmetry and Type: By analyzing the combination of powers in the polynomial, you can determine its symmetry. In our example, \(f(x) = x^3 + x\) is a polynomial of degree 3 (the highest power of \(x\) is 3) plus a linear term, which ultimately tells us this function is odd by its symmetrical properties across the origin.

Polynomial functions are subject to many fascinating properties that provide insights into calculus and algebra.