Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the function is odd, even, or neither. \(f(x)=3 x^{4}\)

Step-by-Step Solution

Answer

The function is even.

Step 1: Determine the Even Function

A function is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \). Let's substitute \( -x \) into the function and see if the condition holds: \[ f(-x) = 3(-x)^4 = 3(x^4) = 3x^4 \] This shows that \( f(-x) = f(x) \), indicating that \( f(x) \) is an even function.

Step 2: Check Odd Function Condition (Optional)

A function is odd if \( f(-x) = -f(x) \). Since we have already determined the function is even, checking for oddness is unnecessary, but for completion: \[ f(-x) = 3(-x)^4 = 3x^4 \] \[ -f(x) = -3x^4 \] Clearly, \( f(-x) eq -f(x) \), confirming it is not odd.

Key Concepts

Function Analysis

Function analysis involves investigating the properties and behavior of functions. This concept aids in understanding whether a function is even, odd, or neither. When tackling such problems, specific definitions help classify the functions:

• **Even Functions**: These have symmetry about the y-axis. For a function \( f(x) \) to be even, it should hold that \( f(-x) = f(x) \) for every \( x \) in its domain.
• **Odd Functions**: They exhibit symmetry about the origin. A function \( f(x) \) is considered odd if \( f(-x) = -f(x) \) for all \( x \) in its domain.
• **Neither Even nor Odd**: If a function doesn't satisfy either condition, it is categorized as neither even nor odd.

Function analysis ensures a deeper understanding of how functions behave based on the changes in the input variable \( x \). By systematically analyzing these aspects, observing symmetry becomes straightforward.

Polynomial Functions

Polynomial functions are algebraic expressions that involve terms in the form \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer. Knowing the power of the highest-degree term in a polynomial forms a critical part of evaluating its properties:

• Polynomials with all even powers tend to be even functions.
• Polynomials where all terms have odd powers are more likely to be odd functions.
• Mixed powers generally yield neither even nor odd functions.

For instance, the function in the problem, \( f(x) = 3x^4 \), is a polynomial function with only one term of even power. Such a structure makes it qualify as an even function. Polynomial functions serve as an excellent arena to apply the principles of function symmetry and determine their classification by simply examining their terms.

Algebraic Properties

Algebraic properties facilitate determining whether a function is even, odd, or neither by manipulating algebraic expressions accordingly. The particular properties of exponents significantly aid in this regard:

• **Even Exponents**: When a negative input is raised to an even exponent, the result is always positive \((-x)^n = x^n\) if \( n \) is even.
• **Odd Exponents**: A negative input raised to an odd exponent results in a negative outcome \((-x)^n = -x^n\) if \( n \) is odd.

These properties clarify why \( f(-x) = f(x) \) for even exponents and \( f(-x) = -f(x) \) for odd exponents. By utilizing these algebraic truths, analyzing whether functions with different power terms are even or odd becomes more intuitive. The example provided reinforces the role of even exponents in affirming the evenness of the function \( f(x) = 3x^4 \).