/*! 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 59 Find the domain of the given fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 59

Find the domain of the given function. Express the domain in interval notation. $$f(x)=2 x-5$$

Short Answer

Expert verified
The domain of \( f(x)=2x-5 \) is \((-, )\).

Step by step solution

01

Understanding the Question

We need to find the domain of the function \( f(x) = 2x - 5 \). The domain of a function consists of all the values of \( x \) for which the function is defined.
02

Identify Constraints

For the given linear function \( f(x) = 2x - 5 \), there are no denominators, square roots of negative numbers, or logarithms involved, which are common constraints. Therefore, there are no restrictions on the values of \( x \).
03

Define the Domain

Since there are no restrictions on \( x \), the domain of \( f(x) = 2x - 5 \) includes all real numbers. In mathematical terms, this is represented as the interval \((-, )\).
04

Express the Domain in Interval Notation

The interval notation expressing all real numbers for the domain is \((-\u001f, )\). This means x can be any real number from negative infinity to positive infinity.

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.

Interval Notation
Interval notation is a shorthand used in mathematics to describe a set of numbers. This can be particularly useful for describing domains of functions.
There are different types of interval notations: closed, open, and half-open.
  • Closed interval: If the endpoints of the interval are included in the set, this is represented with square brackets, e.g., \([a, b]\) means all numbers from \(a\) to \(b\), including \(a\) and \(b\) themselves.
  • Open interval: Uses parentheses to exclude the endpoints. For example, \((a, b)\) represents all numbers between \(a\) and \(b\), but neither \(a\) nor \(b\) is included.
  • Half-open interval: Combines a square bracket and a parenthesis, such as \([a, b)\) or \((a, b]\), where one endpoint is included but not the other.
When describing the domain of a function such as a linear function that spans all real numbers, you would use open intervals to represent extending infinitely in both directions, specified as \((-\infty, \infty)\). This indicates there are no bounds or limits on the values within the domain, as infinity isn鈥檛 an actual number but a concept of endlessness.
Linear Function
A linear function is a basic type of function that creates a straight line when graphed. It has the general form:\[ f(x) = mx + b \] where:
  • \(m\) is the slope, describing the steepness of the line.
  • \(b\) is the y-intercept, indicating where the line crosses the y-axis.
The essential characteristics of linear functions include:
  • Constant Rate of Change: Linear functions have a constant rate of change, so as \(x\) increases by a constant amount, \(f(x)\) does as well.
  • No Curvature: Unlike quadratic or cubic functions, linear functions do not curve. The graph is a straight line.
  • Simple Domain: Typically, linear functions do not have restrictions like square roots or denominators that could limit their domain, which usually includes all real numbers.
The function\( f(x) = 2x - 5 \) is linear because it fits the form of \(mx + b\). Here, the slope \(m = 2\), and the y-intercept \(b = -5\).
Real Numbers
Real numbers encompass virtually all the numbers we encounter in everyday mathematics and are the bedrock of algebra and calculus. Real numbers include:
  • Rational Numbers: These can be expressed as the quotient of two integers, like \(\frac{1}{2}\) or \(3.5\).
  • Irrational Numbers: These cannot be expressed as a simple fraction, such as \(\sqrt{2}\) or \(\pi\).
Real numbers are denoted using the symbol \(\mathbb{R}\) and include:
  • Both positive and negative integers.
  • Fractions and decimals extending infinitely.
  • All infinite decimals, whether repeating or not.
When we talk about the domain of a function like \( f(x) = 2x - 5 \), stating it in terms of all real numbers \( (-\infty, \infty) \) means every value from the set of real numbers works in the function. Real numbers provide a continuous, unbroken spectrum of achievable values for \(x\) without gaps.

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

In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=(5-x)^{1 / 3}, \quad g(x)=5-x^{3}$$

Under what conditions is the linear function \(f(x)=m x+b\) a one-to-one function?

In calculus, the difference quotient \(\frac{f(x+h)-f(x)}{h}\) of a function \(f\) is used to find a new function \(f^{\prime},\) called the derivative off. To find \(f^{\prime},\) we let \(h\) approach \(0, h \rightarrow 0,\) in the difference quotient. For example, if \(f(x)=x^{2}, \frac{f(x+h)-f(x)}{h}=2 x+h,\) and allowing \(h=0,\) we have \(f^{\prime}(x)=2 x\) $$\text { Given } f(x)=\frac{x-5}{x+3}, \text { find } f^{\prime}(x)$$

In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=\frac{x-2}{3}, \quad g(x)=3 x+2$$

Determine whether each statement is true or false. Every odd function is a one-to-one function.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Geometry

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.