Problem 42 - Applied Calculus

Question

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=\frac{x}{x^{2}+9} $$

Step-by-Step Solution

Answer

The function is a rational function.

Step 1: Understanding the Function

First, let's rewrite the given function: \( f(x) = \frac{x}{x^2 + 9} \). This function is in the form of a fraction. The numerator is a polynomial \( x \), and the denominator is a polynomial \( x^2 + 9 \).

Step 2: Identifying the Type of Function

A rational function is a function that can be expressed as the ratio of two polynomials. Since \( f(x) = \frac{x}{x^2 + 9} \) is expressed as a fraction of two polynomials, it is a rational function.

Key Concepts

Polynomial Functions

Polynomial functions are at the core of many mathematical concepts and are widely used in various fields like engineering, physics, and economics. To identify a polynomial function, we look for expressions that involve sums and products of variables raised to a power, where each term of the function is of the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\). Here, \(a_n\) represents a constant coefficient and \(n\) a non-negative integer indicating the term's degree.
For example, \(2x^3 + 3x^2 - x + 5\) is a polynomial of degree 3. The degree of a polynomial function is determined by the highest power of \(x\) present.

• Polynomials are smooth, continuous, and have no breaks in their graphs.
• They do not contain variables in the denominator or any radical signs.

When evaluating the function \(f(x) = \frac{x}{x^2 + 9}\), note that both the numerator, \(x\), and the denominator, \(x^2 + 9\), are polynomials. However, the function itself, because of the division, is not a polynomial but belongs to another important category: rational functions.

Function Identification

In mathematics, identifying a function means determining the type of function based on its algebraic structure. Different types of functions include polynomial, rational, exponential, logarithmic, and piecewise functions.
For function identification:

• Polynomial Functions: Consist of terms made of variables raised to a non-negative integer power.
• Rational Functions: Ratios of two polynomials, such as \(\frac{p(x)}{q(x)}\).
• Exponential Functions: Have a constant base raised to a variable power, like \(a^x\).
• Piecewise Functions: Defined by different expressions based on the input's value range.

To correctly identify \(f(x) = \frac{x}{x^2 + 9}\) as a rational function, observe that it is the ratio of two polynomial expressions, which is the hallmark feature of rational functions.

Mathematics Education

Understanding mathematical concepts thoroughly is essential for effective learning. Mathematics education aims to develop problem-solving skills, logical reasoning, and the ability to make connections between different concepts.
In the context of function identification, it is important to:

• Encourage students to recognize patterns and common structures in mathematical expressions.
• Help learners to distinguish between different types of functions through practice and application.
• Foster a deep understanding of how different functions are constructed and used in problem-solving.

Incorporating exercises that involve the identification of function types, like the given example \(f(x) = \frac{x}{x^2 + 9}\), can significantly enhance a student’s ability to understand and apply different mathematical concepts.
Education techniques that blend theoretical principles with practical applications can increase retention and facilitate a deeper understanding of mathematics. Such practices ensure learners not only memorize function types but also appreciate their practical uses and implications in various contexts.