/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 For Problems \(15-30\), specify ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 30

For Problems \(15-30\), specify the domain for each function. (Objective 3) $$ y=-2 x^{4} $$

Short Answer

Expert verified
The domain of \( y = -2x^4 \) is \( (-\infty, \infty) \).

Step by step solution

01

Understand the Nature of the Function

The given function is a polynomial function: \( y = -2x^4 \). Polynomial functions are defined for all real numbers because there are no restrictions such as division by zero or taking the square root of a negative number.
02

Identify any Restrictions

Since the function \( y = -2x^4 \) is a polynomial, there are no restrictions on the domain. Polynomial functions can take any real number as input.
03

Specify the Domain

Since there are no restrictions, the domain of the function is all real numbers. We denote the domain as \( (-\infty, \infty) \). This means that as \( x \) ranges over all real numbers, the function \( y = -2x^4 \) is defined.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Polynomial Function
Polynomial functions are a fundamental building block in mathematics. They are expressions involving variables raised to whole number exponents and multiplied by coefficients. A polynomial function takes the general form:
  • Linear: For example, \( y = ax + b \)
  • Quadratic: Such as \( y = ax^2 + bx + c \)
  • Higher degrees: Like \( y = ax^n + bx^{n-1} + \ldots \)
In the given exercise, \( y = -2x^4 \) is a polynomial of degree 4, where 4 is the highest exponent. This specific form doesn't involve division by a variable or square roots, which means it doesn't impose any restrictions on its inputs.
Because of their simple structure, polynomial functions are defined for all real numbers, making their domain very broad and inclusive.
Real Numbers
In mathematics, real numbers include all numbers that can be found on the number line. This wide category encompasses several types of numbers, such as:
  • Rational numbers: Numbers that can be expressed as fractions, like \( \frac{1}{2} \) or \( -3 \)
  • Irrational numbers: Numbers that cannot be expressed as a simple fraction, like \( \pi \) or \( \sqrt{2} \)
  • Integers: Whole numbers, both positive, negative, and zero
When we say a function is defined for "all real numbers," we mean it can take any of these types of numbers as inputs. Each specific input results in a corresponding output based on the function's rule. The domain of a polynomial function like \( y = -2x^4 \) is therefore the entire set of real numbers since there are no restrictions on the types of numbers it can accept.
Mathematical Notation
Mathematical notation is a system of symbolic representations and a universal language among mathematicians. It allows for concise, clear, and precise expression of mathematical ideas. Here's how it applies to our exercise:
  • Function notation: \( y = -2x^4 \) describes a relationship where \( y \) depends on \( x \)
  • Domain notation: \( (-\infty, \infty) \) suggests that \( x \) takes any value from negative infinity to positive infinity. The use of parentheses \( ( \) and \( ) \) denotes that these endpoints are not included, which aligns with the concept of infinity
This exercise illustrates how notation simplifies complex ideas, helping to communicate broader concepts such as domains and functions in a compact form
Understanding these notations are key to navigating and solving mathematical problems effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve each of the following quadratic equations, and check your solutions. $$(x+3)^{2}=-16$$

Solve each of the following quadratic equations, and check your solutions. $$y^{2}-2 y=-19$$

The ABC Car Rental uses the function \(f(x)=26\) for the charges, in dollars, for the daily use of a car driven up to and including 200 miles. When a car is driven more than 200 miles per day, they use the function \(g(x)=26+0.15(x-200)\) to determine the charges. How much would they charge for daily driving of 150 miles, 230 miles, 360 miles, and 430 miles?

Solve each of the following quadratic equations, and check your solutions. $$x^{2}-4 x+7=0$$

The function \(A(s)=s^{2}\) expresses the area of a square as a function of the length of a side \(s\) of the square. Compute \(A(3), A(17), A(8.5), A(20.75)\), and \(A(11.25)\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.