Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Find the domain of the expression.\(3 x^{2}-4 x+7\)

Step-by-Step Solution

Answer

The domain of the expression \(3x^2 - 4x + 7\) is all real numbers.

Step 1: Identify restrictions

Find values of \(x\) where the function is undefined (division by zero, negative under even roots, etc.).

Step 2: State the domain

The domain is The domain of the expression \(3x^2 - 4x + 7\) is all real numbers..

Key Concepts

Polynomial Function

Let's begin by understanding what a polynomial function is. At its core, a polynomial function is an algebraic expression made up of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A standard form of a polynomial function is expressed as \(a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{2}x^{2} + a_{1}x + a_{0}\), where:\
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• \(a_{n}, a_{n-1}, ..., a_{0}\) are constants called coefficients,\
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• \(x\) is the variable,\
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• \(n\) is a non-negative integer that represents the degree of the polynomial.\
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Polynomial functions are continuous and smooth curves without breaks or sharp corners. They are also unbounded, meaning they continue indefinitely in the positive and negative directions along the y-axis. A key property of polynomial functions is that they have a domain of all real numbers, which leads us into the concept of evaluating their domain.

Real Numbers

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions, as we've mentioned, this includes all real numbers. Real numbers are the collection of all the rational and irrational numbers. This set contains all the numbers we typically use in everyday arithmetic:\
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• Rational numbers, which can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\) is not zero,\
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• Irrational numbers, which cannot be written as a simple fraction and have non-repeating, non-terminating decimal parts, such as \(\pi\) and \(\sqrt{2}\).\
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In algebra, when we say a function's domain is all real numbers, we imply that there is nothing to restrict the value of the input variable x. This is typically the case for polynomial functions, unless stated otherwise in the context of the problem. Hence, they can be evaluated over the entire number line without any limitations.

Algebraic Expressions

Central to our understanding of polynomial functions is the concept of algebraic expressions. An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(x\) or \(y\)), and operators (such as add, subtract, multiply, and divide). The function \(3 x^{2}-4 x+7\) from our exercise is an example of an algebraic expression called a trinomial, meaning it has three terms.
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Algebraic expressions are significant because they represent quantities without fixed values, often unknown or variable. They are like a recipe that describes how to perform a set of instructions depending on input values. When it comes to polynomial functions, algebraic expressions allow us to present relationships between variables in a way that can be easily analyzed and manipulated for finding solutions to various problems, including the determination of domains.