Problem 43 - 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.) $$ f(x)=3 x^{2}-2 x $$

Step-by-Step Solution

Answer

The function \( f(x)=3x^{2}-2x \) is a polynomial function.

Step 1: Understanding Polynomials

A polynomial function is a mathematical expression involving a sum of powers of one or more variables multiplied by coefficients. The general form is \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where the coefficients \( a_i \) are real numbers and \( n \) is a non-negative integer. There should be no division by variables or other more complex functions like square roots or logarithms.

Step 2: Analyzing the Given Function

The given function is \( f(x)=3x^{2}-2x \). Notice that this function consists of terms with the variable \( x \) raised to non-negative integer powers — in this case, the powers are 2 and 1. The coefficients (3 and -2) are real numbers, and there are no divisions by \( x \) or other non-polynomial operations.

Step 3: Identifying Function Type

Since \( f(x)=3x^{2}-2x \) matches the form of a polynomial function where the powers of \( x \) are non-negative integers and all coefficients are real numbers, it can be classified as a polynomial function.

Key Concepts

Polynomial Identification

A great starting point in understanding mathematical functions is to identify whether they are polynomials. Polynomials are one of the most fundamental types of mathematical expressions. They consist of terms where variables are raised to whole number powers, and these terms are summed together to form the expression. For a function to be polynomial, it must adhere to a specific structure: no division by a variable, no complex operations like square roots, and each term is a product of a coefficient and a power of a variable.

Let's take the function \( f(x) = 3x^2 - 2x \). Here, both terms \(3x^2\) and \(-2x\) fit the criteria of polynomials: they have coefficients (3 and -2 respectively) and the powers of \(x\) are non-negative integers (2 and 1).

• Coefficients: Real numbers like 3 and -2.
• Powers: Non-negative integers (2, 1).
• No divisors: No division by \(x\) or any variable.

By meeting these conditions, we can confidently classify \( f(x) = 3x^2 - 2x \) as a polynomial function.

Function Types

Mathematical functions come in various types, each with its unique properties and usages. Recognizing these types is crucial for solving equations and applying the right mathematical rules. Most common types include polynomial functions, rational functions, exponential functions, and piecewise linear functions.

• Polynomial Functions: These involve terms of variables raised to whole number powers. They are used in modeling real-world situations such as motion and growth.
• Rational Functions: These are expressed as the ratio of two polynomial functions, essentially a fraction of polynomials.
• Exponential Functions: The variable is in the exponent, e.g., \(f(x) = a b^x\).
• Piecewise Linear Functions: These functions are defined by different linear equations over different intervals of the domain.

By checking the form and operations within a function, one can ascertain its type. For example, \( f(x) = 3x^2 - 2x \), due to its structure, falls snugly into the polynomial category.

Mathematical Expressions

Understanding mathematical expressions is like learning a new language that involves symbols and numbers instead of words. Expressions are the building blocks of equations and functions, describing a combination of numbers, variables, and operations. In every expression, the arrangement and interaction of variables with coefficients and operations define its type and characteristics.

In the expression \( f(x) = 3x^2 - 2x \):

• Variables: In this case, we have \(x\), which can take various values.
• Coefficients: The numbers 3 and -2, which scale the terms they multiply.
• Operations: Involves multiplication (e.g., \(3 \times x^2\)) and subtraction (between terms).
• Structure: The polynomial structure ensures the expression comprises terms where variables are raised to non-negative powers.

Understanding these elements not only helps in correctly identifying the type of function but also in performing mathematical operations and problem-solving effectively.