Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=3 x^{5}-x^{3}+7 x$$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Understand Even and Odd Functions

A function is even if it satisfies \(f(-x) = f(x)\) for all \(x\), and it is odd if \(f(-x) = -f(x)\). First, we'll compute \(f(-x)\) and then determine which of these properties apply.

Step 2: Calculate \(f(-x)\)

Substitute \(-x\) into the function: \( f(-x) = 3(-x)^5 - (-x)^3 + 7(-x) \). Simplify each term: \((-x)^5 = -x^5\), \((-x)^3 = -x^3\), and \(7(-x) = -7x\). So, \( f(-x) = 3(-x^5) - (-x^3) + (-7x) = -3x^5 + x^3 - 7x \).

Step 3: Compare \(f(-x)\) with \(-f(x)\)

The original function is \(f(x) = 3x^5 - x^3 + 7x\). To find \(-f(x)\), multiply each term by \(-1\): \-f(x) = -3x^5 + x^3 - 7x\. We observe that \f(-x) = -3x^5 + x^3 - 7x\ matches \-f(x)\. Thus, \(f(x)\) satisfies the condition for odd functions.

Step 4: Conclusion

Since \(f(-x) = -f(x)\), the function \(f(x) = 3x^5 - x^3 + 7x\) is an odd function.

Key Concepts

Even and Odd Functions

Functions can be categorized based on their symmetry properties, and two common types are **even** and **odd functions**. Understanding these properties helps in analyzing the behavior of functions visually and algebraically.

• Even Functions: These are symmetric about the y-axis. Mathematically, a function is considered even if it satisfies the condition \(f(-x) = f(x)\) for all inputs \(x\). Examples include functions like \(x^2\) and \(\cos(x)\).
• Odd Functions: These are symmetric about the origin. A function is classified as odd if \(f(-x) = -f(x)\). This means that if you rotate the graph 180 degrees around the origin, it looks the same. Examples include \(x^3\) and \(\sin(x)\).

Understanding these properties allows you to graph and predict the behavior of functions without performing extensive calculations every time.

Polynomial Functions

Polynomial functions are a central topic in algebra and calculus. They are expressions consisting of variables raised to various powers and multiplied by coefficients. For instance, \(f(x) = 3x^5 - x^3 + 7x\) is a polynomial function.

• Structure: Polynomial functions have the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where the highest power of \(x\) is called the degree of the polynomial. The degree indicates the most significant impact on the function's shape and behavior.
• Behavior: The degree of the polynomial dictates end-behavior. For instance, polynomials of even degree have similar rise and fall characteristics, while odd-degree polynomials exhibit different end behaviors at each side of the graph.

Moreover, the process of determining if a polynomial is even or odd involves checking the exponents of the terms. If all powers are even, the polynomial is even. If all powers are odd, it could be odd, considering the coefficients as well.

Function Symmetry

Symmetry in functions is a fascinating concept that offers insight into the graphical representation and properties of functions.

• Y-Axis Symmetry (Even Functions): The graph appears unchanged if reflected over the y-axis. This occurs if \(f(-x) = f(x)\). It signifies an equal shape on either side of the y-axis.
• Origin Symmetry (Odd Functions): If rotating the graph 180 degrees about the origin leaves it unchanged, it has origin symmetry, fulfilling \(f(-x) = -f(x)\). This type of symmetry is visually intriguing as it shows rotational symmetry rather than reflection.
• No Symmetry: Some functions do not exhibit either y-axis or origin symmetry. These functions can still be graphed, but without a specific symmetry attribute.

Mastering these symmetry properties helps in understanding the fundamental nature and application of various mathematical functions, making it easier to solve equations and visualize transformations.