Problem 49 - Calculus: One and Several Variables

Question

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

Step-by-Step Solution

Answer

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

Step 1: Test for even function

An even function is defined as \( f(x) = f(-x) \). Substitute -x into the function \( f(x) = x^{3} \), we get \( f(-x) = (-x)^{3} \) which simplifies to \( -x^{3} \). As \( f(-x) \) isn't equal to \( f(x) \), this function is not even.

Step 2: Test for odd function

An odd function is defined as \( f(-x) = -f(x) \). From step 1, we've seen that \( f(-x) = -x^{3} \), and if we substitute \( f(x) \), we see that \( -x^{3} = - f(x) \). So, the function \( f(x) = x^{3} \) is odd since \( f(-x) = -f(x) \).

Key Concepts

Polynomial Functions

Polynomial functions are an important class of functions that have variables raised to whole number exponents. These functions are expressed in the general form \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]where each coefficient \(a_i\) is a real number, and \(n\) is a non-negative integer representing the highest degree of the polynomial. The degree of a polynomial indicates the highest power of the variable present in the expression, making it crucial for determining the behavior of the function.

Characteristics of polynomial functions include smooth and continuous graphs, without breaks or sharp corners. They also can be classified according to their degree, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on. In our example, the function \(f(x) = x^3\) is a cubic polynomial, making it interesting because cubic functions can display more complex behaviors and shapes compared to linear or quadratic functions.

Understanding polynomial functions helps in a variety of applications, from modeling natural phenomena to solving engineering problems. When working with these functions, identifying their properties can simplify analysis and solution finding.

Function Properties

Function properties help us make sense of the behavior and characteristics of functions. Some essential properties to look out for include continuity, differentiability, and symmetry.

• Continuity: A function is continuous if it can be drawn without lifting the pen from the paper. Polynomial functions are always continuous.
• Differentiability: Differentiable functions have derivatives everywhere within their domain. All polynomial functions are differentiable throughout their domain since they have smooth curves.
• Symmetry: This is a crucial property when identifying whether a function is even or odd. Symmetry helps us understand the visual representation of functions.

Specifically, symmetry in respect to the y-axis or the origin tells us about even and odd functions, respectively. Recalling these properties enhances our comprehension of functions and their graphs, making it easier to manipulate and use in various contexts.

Symmetry in Functions

Symmetry in functions is a fascinating aspect as it provides a deeper insight into the nature of the function's graph. We categorize functions into even, odd, or neither based on their symmetry.

• Even Functions: These functions satisfy the condition \(f(x) = f(-x)\). A graph of an even function is symmetric about the y-axis. Examples include \(f(x) = x^2\) or \(f(x) = \,\cos(x)\).
• Odd Functions: Odd functions meet the condition \(f(-x) = -f(x)\). They exhibit symmetry about the origin. Our original function \(f(x) = x^3\) is a classic example of an odd function.
• Neither Even nor Odd: Some functions don't fit neatly into either category, lacking specific symmetrical properties. These require separate analysis.

Recognizing symmetry simplifies the process of graphing these functions and can help in predicting their behavior. By breaking down even and odd functions, students can more easily grasp how changes to input values affect outputs, aiding in problem-solving and graph interpretation.