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Chapter 1: Problem 5

Find the domain of the expression. $$4 x^{2}-10 x+3$$

Short Answer

Expert verified
The domain of the expression is all real numbers, \((-\infty, \infty)\).

Step by step solution

01

Understand the Expression Type

The given expression is a quadratic polynomial, which is generally represented in the form \( ax^2 + bx + c \). Here, it is \( 4x^2 - 10x + 3 \).
02

Recognize Domain Characteristics of Polynomials

Polynomials, like the given quadratic expression, are defined for all real numbers. This means there are no restrictions on the values that \( x \) can take.
03

Conclude the Domain

Since the quadratic expression is defined for all real numbers, the domain is all real numbers, which can be written in interval notation as \((-\infty, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Polynomials
Quadratic polynomials are fundamental concepts in algebra. These polynomials are expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The presence of the \(x^2\) term makes it quadratic. This means the highest degree of the variable \(x\) in the equation is 2.
Quadratic polynomials appear in various mathematical models, describing parabolic shapes. This is because their graphs are parabolas.
  • The coefficient \(a\) determines the direction of the parabola (upwards or downwards).
  • The vertex of the parabola gives the maximum or minimum point, depending on the parabola's direction.
  • Quadratic polynomials can be solved by finding the roots, which are the solutions of \(ax^2 + bx + c = 0\).
Real Numbers
Real numbers encompass all the numbers we use in everyday mathematics. They include rational numbers like fractions and decimals, as well as irrational numbers, such as \(\pi\) and \sqrt{2}. In other words, the set of real numbers includes all the numbers that lie on the number line.
When dealing with domains of polynomials, like the quadratic polynomial in our example, it's important to know that these functions are generally defined for all real numbers.
  • Rational numbers include numbers like \(\frac{1}{2}\) and -3.
  • Irrational numbers are those that cannot be written as simple fractions, such as \(\pi\) and \(e\).
  • Negative numbers, zero, positive numbers, integers, fractions, and decimals all make up the real number line.
Interval Notation
Interval notation is a way to describe sets of numbers in a concise form. It is often used when expressing domains or ranges of functions. Instead of listing all numbers between two values, interval notation uses brackets and parentheses to indicate which sections of the number line are included.
  • Parentheses \(( )\) indicate that an endpoint is not included in the interval.
  • Brackets \([ ]\) indicate that an endpoint is included in the interval.
  • The symbol \(-\infty\) is always accompanied by a parenthesis, as infinity is not a specific number but a concept.
For example, the interval notation \((-\infty, \infty)\) specifies all real numbers from negative infinity to positive infinity, indicating no restrictions on the domain of the function.

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Most popular questions from this chapter

Factor the expression completely. $$2 x^{2}+7 x-4$$

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$(x-1)^{7 / 2}-(x-1)^{3 / 2}$$

Factor the expression completely. $$\left(a^{2}+1\right)^{2}-7\left(a^{2}+1\right)+10$$

Perform the indicated operations and simplify. $$y^{1 / 3}\left(y^{2 / 3}+y^{5 / 3}\right)$$

Factor the expression completely. (This type of expression arises in calculus when using the "Product Rule.") $$3(2 x-1)^{2}(2)(x+3)^{1 / 2}+(2 x-1)^{3}\left(\frac{1}{2}\right)(x+3)^{-1 / 2}$$
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