Problem 1 - Algebra 2 and Trigonometry

Question

Is the function \(\mathrm{f}=\\{(x, y) : x y=20\\}\) a polynomial function? Explain why or why not.

Step-by-Step Solution

Answer

No, it is not a polynomial function because it involves a negative exponent.

Step 1: Understand the Given Function

The given function is a set of ordered pairs \((x, y)\) such that their product \(xy = 20\). This relationship defines \(y\) in terms of \(x\) within the set defined by this equation.

Step 2: Identify Characteristics of a Polynomial Function

A polynomial function is an expression involving sums of powers of the variable, with non-negative integer exponents, and coefficients taken from a specific field (usually real numbers). An example of a polynomial function is \(f(x) = ax^n + bx^{n-1} + ... + c\), where \(n\) is a non-negative integer.

Step 3: Relate Given Function to Polynomial Definition

The expression \(xy = 20\) can be rearranged to solve for \(y\), giving \(y = \frac{20}{x}\). The expression \(\frac{20}{x}\) is equivalent to \(20x^{-1}\). Recall that polynomial functions do not have negative exponents for any term involving the variable.

Step 4: Conclude Based on Analysis

Since the function \(y = 20x^{-1}\) involves a term with a negative exponent, it does not meet the definition of a polynomial function, which requires non-negative integer exponents.

Key Concepts

Negative Exponents

In mathematics, negative exponents can often appear confusing at first, but they're quite straightforward once you know the basic rules. An exponent refers to the number of times a number is multiplied by itself. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, consider the expression \(x^{-n}\), where \(x\) is the base and \(-n\) is the exponent. This is equivalent to \(\frac{1}{x^n}\).
Let's break this down:

• The negative sign in the exponent means you take the reciprocal of the base.
• The exponent tells you how many times to multiply the reciprocal base by itself.

For a concrete example, let's use \(x^{-2}\). This is the same as \(\frac{1}{x^2}\), which means you divide 1 by \(x\) multiplied by itself twice. Understanding this concept is crucial when analyzing functions that could involve terms with negative exponents, especially in distinguishing polynomial functions from other types.

Non-Negative Integer Exponents

A polynomial function is defined by the lack of negative exponents in any terms involving the variables. Instead, each variable in a polynomial is raised to a non-negative integer exponent. Typical terms in a polynomial function look like \(ax^n + bx^{n-1} + \, ... \, + c\), where \(a, b,\) and \(c\) are coefficients, and each power \(n\) is a non-negative integer (0, 1, 2, etc.).
Let's explore this further:

• Non-negative means that the exponent is either zero or a positive number.
• Integer signifies that the exponent is a whole number, so fractions are not allowed.

This requirement is essential to the definition of polynomial functions because they are built on the standard operations of addition, subtraction, and multiplication of real numbers, which naturally restrict exponents in this way. Remember, constants can be expressed as terms in the polynomial with exponents of zero, such as \(c = c \cdot x^0\). Understanding non-negative integer exponents helps to classify and construct polynomial functions correctly.

Function Definition

When we talk about functions in mathematics, we're referring to relationships between sets of inputs and outputs, where each input is related to exactly one output. In terms of polynomial functions, this relationship is expressed using variables raised to non-negative integer exponents.
Several key points characterize functions:

• A function maps each input to precisely one output.
• It can be represented as graphical, symbolic, or numerical relationships.
• The notation \(f(x)\) signifies that \(f\) is a function of \(x\).

Specifically, a polynomial function is a function that can be expressed in the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\), where \(a_n, a_{n-1}, \, ..., \) and \(a_0\) are constants. The defining features involve non-negative integer exponents and the assurance that the function will not have any undefined points or asymptotic behavior, making them easier to analyze compared to other types of functions. Recognizing a polynomial function involves ensuring every component adheres to these guidelines.