Problem 46 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is a linear or nonlinear function. If \(f\) is a linear function, determine if \(f\) is a constant function. Support your answer by graphing \(f\). $$ f(x)=x^{3} $$

Step-by-Step Solution

Answer

The function \( f(x) = x^3 \) is nonlinear and not a constant function.

Step 1: Recognize the form of the function

Identify the highest power of the variable in the function expression. The given function is \( f(x) = x^3 \). Here, the degree of the polynomial is 3, as the highest power of \( x \) is 3.

Step 2: Classify the function based on its degree

A function is linear if it can be written in the form \( f(x) = ax + b \), where \( a eq 0 \). The presence of any power of \( x \) greater than 1 indicates a nonlinear function. Since \( f(x) = x^3 \) involves \( x^3 \), it is not a linear function.

Step 3: Determine if it's a constant function

A constant function is one where the function value does not change regardless of \( x \), typically expressed \( f(x) = c \), where \( c \) is a constant. \( f(x) = x^3 \) clearly changes with different values of \( x \), so it's not a constant function.

Step 4: Graph the function to support the analysis

Graph the function \( f(x) = x^3 \). The graph of \( x^3 \) is a smooth curve passing through the origin and is symmetric with respect to the origin. This curvature in the graph further confirms that the function is nonlinear, as a linear graph would be a straight line.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that are composed of variables and coefficients, connected using addition, subtraction, and multiplication. They do not involve division by variables or exponentiation with variables in the exponent. Polynomials look like a sum of multiple terms, each with a constant multiplied by a power of one or more variables. Here are some key points to remember about polynomial functions:

• A term in a polynomial consists of a coefficient and a variable raised to a non-negative integer power.
• The simplest form is a monomial, which has just one term, like \( x^3 \).
• Polynomials are classified based on the number of terms: a binomial has two terms, and a trinomial has three.

In the given exercise, the function \( f(x) = x^3 \) is a monomial because it only comprises one term. It is essential to recognize this to determine the nature of the function as linear or nonlinear.

Degree of a Polynomial

The degree of a polynomial is one of the most crucial attributes to understand its behavior and to classify the function. The degree refers to the highest power of the variable in the polynomial expression. This power determines several aspects of the polynomial's functionality:

• The degree tells us about the maximum number of roots or solutions the polynomial might have.
• It also indicates the number of times the graph of the polynomial can intersect the x-axis.
• Polynomials of degree 1 are linear, while those of higher degrees, like degree 2 or more, are nonlinear.

In \( f(x) = x^3 \), the degree is 3. Since the degree is greater than 1, \( f(x) \) is classified as a nonlinear function. It is essential because identifying the degree helps predict the shape and behavior of the graph.

Graphing Functions

Graphing functions is a visual method for understanding the behavior and classification of functions. A graph of a function can instantly reveal certain characteristics that are not as easily noticeable in the algebraic form.

When graphing a polynomial like \( f(x) = x^3 \), it is helpful to plot several points by substituting values into the function and calculating the results. The graph of \( x^3 \) has special properties:

• It passes through the origin (0, 0), meaning \( f(0) = 0^3 = 0 \).
• It is symmetric about the origin, demonstrating that it is an odd function.
• The curve gives a visual representation of the function's nonlinear nature, as it is not a straight line.

Thus, the graph of \( x^3 \) not only reinforces the function's nonlinearity but also helps confirm the degree status visually. Understanding the graph provides insights into the function's overall characteristics and the polynomial's nature in contexts like growth and behavior dynamics.