/*! 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 16 Find the domain of the function.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 16

Find the domain of the function. $$ f(x)=x^{6}-\sqrt{2} x^{3}-\pi $$

Short Answer

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

Step by step solution

01

Understand the Problem

The domain of a function represents all the possible input values (usually 'x' values) for which the function is defined. The function given is a polynomial function.
02

Identify the Type of Function

The function provided is \[f(x) = x^{6} - \sqrt{2} x^{3} - \pi\]This is a polynomial function because it is composed of terms that contain variables raised to whole number exponents.
03

Determine the Domain of a Polynomial Function

For any polynomial function, the domain is all real numbers because the function is defined for every real value of \(x\). There are no restrictions such as division by zero or square roots of negative numbers in this function.
04

Present the Domain

The domain for \[f(x) = x^{6} - \sqrt{2} x^{3} - \pi\]is all real numbers, which can be denoted as \((-\infty, \infty)\).

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
A polynomial function is an expression made up of variables, coefficients, and non-negative integer exponents. This type of function is important in algebra and calculus due to its well-behaved characteristics and simplicity. The form of a polynomial function is typically:
  • The general form is \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are coefficients.
  • The highest exponent of the variable \(x\), which is \(n\), is called the degree of the polynomial.
  • Each term is called a "monomial", and the degree determines the behavior of the polynomial graph at extreme values of \(x\).
In the original exercise, the function \[f(x) = x^{6} - \sqrt{2} x^{3} - \pi\]is a polynomial because it consists of terms where the variable \(x\) is raised to whole number powers.
Real Numbers
Real numbers consist of all the numbers that can be found on the number line. This includes both rational numbers, such as integers and fractions, and irrational numbers, which cannot be expressed as simple fractions. Real numbers are very important in defining domains of functions because they represent possible values that include:
  • Positive numbers like 3 and 5.6.
  • Negative numbers like -2 and -3.5.
  • Zero, which is neither positive nor negative.
  • Irrational numbers like \(\pi\) and \(\sqrt{2}\).
Understanding real numbers is crucial when working on the domain of functions, as seen in the original exercise, where the domain is all real numbers, meaning any real number can be used as an input \(x\) value for the function.
Function Definition
A function is a special relationship where each input is related to exactly one output. Functions can be represented in various forms, such as equations, graphs, or tables. The key characteristics of function definition include:
  • Domain: The set of all possible input values (usually 'x' values). For the exercise function \[f(x) = x^{6} - \sqrt{2} x^{3} - \pi\], the domain is all real numbers.
  • Codomain: The set of potential outputs that include all possible values the function can take.
  • Range: The actual set of outputs the function produces.
Understanding these basic parts helps one in analyzing and working with different types of functions. In our case, the function is a polynomial, making its domain simple and unrestricted鈥攕panning all real numbers.

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

Sketch the region in the plane satisfying the given conditions. \(x>2\) and \(y<1\)

Starting at noon, \(A\) flies 2400 miles from New York to San Francisco at a velocity of 400 miles per hour. \(B\) starts the same trip at \(2: 00\) P.M. the same day with a velocity of 800 miles per hour. Express the distance \(D\) between \(A\) and \(B\) at any instant between noon and \(5: 00 \mathrm{P} . \mathrm{M}\). in terms of the time in hours elapsed after noon.

a. Describe the region consisting of all \((x, y)\) for which $$ (x, y)=(x,-y) . $$ b. Describe the region consisting of all \((x, y)\) for which $$ (x, y)=(-x, y) . $$

Solve the equation. $$ |x|=\pi $$

Let \(f\) be a function, and let \(g(x)=f(x)-3 .\) What is the relationship between the graphs of \(f\) and \(g\) ?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.