Problem 77 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understand Even and Odd Functions

An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \) in the function's domain. An odd function satisfies \( f(-x) = -f(x) \) for all \( x \) in the domain. If a function does not meet either condition, it is neither even nor odd.

Step 2: Substitute \(-x\) into the function

Replace \( x \) with \( -x \) in the given function \( f(x) = x^3 + \cos x \) to get \( f(-x) = (-x)^3 + \cos(-x) \).

Step 3: Simplify \( f(-x) \)

Calculate \( f(-x) \). We know \((-x)^3 = -x^3\) and \(\cos(-x) = \cos x\) due to cosine's even property.Thus, \( f(-x) = -x^3 + \cos x \).

Step 4: Compare \( f(x) \) and \( f(-x) \)

Compare the expressions:- \( f(x) = x^3 + \cos x \)- \( f(-x) = -x^3 + \cos x \)Since \( f(-x) eq f(x) \) and \( f(-x) eq -f(x) \), the function is neither even nor odd.

Key Concepts

Function Analysis

When analyzing functions to determine their properties, it's crucial to understand the concepts of even and odd functions. These categorizations provide insights into a function's symmetry:

• **Even Functions:** These functions are symmetric about the y-axis. If you replace every instance of \(x\) with \(-x\), the function \(f(-x)\) will be identical to \(f(x)\). A visual example is the parabola \(y = x^2\).
• **Odd Functions:** Unlike even functions, odd functions are symmetric about the origin. For every \(x\), the function satisfies \(f(-x) = -f(x)\). A simple visual cue is the curve \(y = x^3\).
• **Neither:** If a function doesn't conform to either rule above, it is classified as neither even nor odd. Such functions don't have a standard symmetry property.

Understanding these categories helps you quickly identify a function's behavior, making it easier to predict how it reacts when variables change. It is always helpful to perform substitutions like \(f(x)\) and \(f(-x)\) to confirm the function's class.

Cosine Function Properties

The cosine function, often written as \(\cos x\), is a fundamental trigonometric function exhibiting several unique properties. One of these is its inherent evenness:

• **Even Nature:** The cosine function satisfies \(\cos(-x) = \cos(x)\) for all \(x\). This means it reflects perfectly around the y-axis. In terms of the unit circle, cosine represents the horizontal coordinate of a point.
• **Periodicity:** Cosine functions are periodic with a period of \(2\pi\), meaning they repeat their values in this interval continuously across the x-axis.
• **Amplitude and Range:** With an amplitude of 1, the range of \(\cos x\) is \([-1, 1]\).

Understanding these attributes of cosine aids in simplifying problems, particularly when analyzing combinations of different functions, like \(f(x) = x^3 + \cos x\). Here, the properties of cosine simplify the analysis of function symmetry. The even nature of cosine helps directly compare \(f(x)\) and \(f(-x)\).

Polynomial Functions

Polynomial functions, such as \(x^3\), come in various forms, distinguished by their degree—dictated by the highest power of \(x\). The degree of a polynomial determines its fundamental behavior:

• **Degree Characteristics:** A polynomial's degree reveals its symmetry. Odd-degree polynomials, like \(x^3\), tend to be odd functions when standing alone, meaning they generally reflect through the origin.
• **Terms and Coefficients:** These functions consist of terms like \(ax^n\), where \(a\) is the coefficient. They guide the polynomial’s weight or "influence" for different powers of \(x\).
• **Behavior and Graph:** For \(x^3\), the function starts from negative and rises to positive (increasing throughout), illustrating odd function symmetry.

In the context of combined functions such as \(f(x) = x^3 + \cos x\), analyzing polynomial parts alone can be insightful. The presence of \(x^3\) tries to push the function into odd classification, but combined with an even function like cosine, results in a function that’s neither even nor odd overall. Understanding the composition of polynomials and their degree helps predict real-world applications and function interactions.