Problem 60 - A Graphical Approach to College Algebra

Question

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=x^{3}-4 x$$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Understand the Definitions

A function is **even** if \( f(-x) = f(x) \) for all \( x \), which means it is symmetric about the y-axis. A function is **odd** if \( f(-x) = -f(x) \) for all \( x \), indicating it has rotational symmetry about the origin.

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

Substitute \(-x\) into the function \( f(x) = x^3 - 4x \). This gives us:\[f(-x) = (-x)^3 - 4(-x) = -x^3 + 4x.\]

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

Now compare \( f(-x) = -x^3 + 4x \) with \( f(x) = x^3 - 4x \). To test if the function is even, we check if \( f(-x) = f(x) \), which isn't true here.To test if it's odd, compare \( f(-x) = -x^3 + 4x \) to \(-f(x) = -(x^3 - 4x) = -x^3 + 4x \). They are equal, thus the function is odd.

Key Concepts

Function Symmetry

Function symmetry is a fundamental concept in understanding the behavior of functions graphically and algebraically. Symmetrical functions around the y-axis are called even functions, while those with origin symmetry are termed odd functions. This property makes it easier to predict points on the graph if one side is known.
Symmetry in functions can be checked using division rules:

• An **even function** remains unchanged if you replace each instance of the variable with its negative, i.e., if for all values of x, replacing x with -x gives you the same function value. Mathematically, this is represented as \( f(-x) = f(x) \).
• An **odd function** will mirror itself about the origin if the variable is replaced by its negative counterpart and the result is the negative of the function's original value, given by \( f(-x) = -f(x) \).

Observing symmetries in polynomial functions is particularly straightforward as these properties often correspond to specific exponents, aiding in their thorough classification.

Polynomial Functions

A polynomial function is a mathematical expression that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of a variable. Polynomial functions form a very important class of functions because they are used widely in calculus and over various fields of science and engineering.
Polynomials are classified based on their degree:

• The **degree** of a polynomial is defined as the highest power of the variable in the polynomial. For instance, the polynomial \(x^3 - 4x\) is of degree 3, which is also called a cubic polynomial.
• The **leading coefficient** is the coefficient of the term with the highest power. In the expression \(x^3 - 4x\), the leading coefficient is 1.

Such polynomial functions are pivotal when analyzing function behaviors like symmetry, as their terms can distinctly show even or odd properties. Remember, analyzing each term separately can help in identifying the overall symmetry of the polynomial function.

Function Evaluation

Function evaluation is the process of substituting specific values into a function to determine the function's output. It is akin to plugging in numbers to see what the calculation or expression gives back.
In terms of function symmetry, function evaluation allows us to explore the behavior of functions for values and their negatives effectively. Here is how you approach it:

• First, substitute \(-x\) into the function and compute what it yields. This is a straightforward algebraic manipulation like the example \(f(-x) = (-x)^3 - 4(-x)\), resulting in \(-x^3 + 4x\).
• Secondly, compare this resulting expression with the original function \(f(x)\). If they match or negate completely, you can conclude the function's symmetry.

This method is especially useful in determining the characteristics of polynomial functions and identifying even or odd functions through comparison.