/*! 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 81 Let \(f(x)=3 x+1\) and \(g(x)=x^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 81

Let \(f(x)=3 x+1\) and \(g(x)=x^{2}-2 .\) Find the following. The domain of \(f\)

Short Answer

Expert verified
The domain of \(f(x) = 3x + 1\) is \((-fty, +fty)\).

Step by step solution

01

Understand the function

The function given is a linear function: \(f(x) = 3x + 1\). It is important to identify the type of function in order to determine its domain.
02

Identify the form of the function

Since \(f(x) = 3x + 1\) is a linear function, it does not have any restrictions like square roots or denominators that could make the function undefined.
03

Determine the domain of a linear function

For a linear function like \(f(x) = 3x + 1\), the domain is all real numbers because there are no values of \(x\) that will make the function undefined.
04

Write the domain in interval notation

Since the domain is all real numbers, it can be written in interval notation as \((-fty, +fty)\).

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.

linear function
A linear function is one of the most straightforward types of functions you'll encounter in math. Linear functions have the form: \[ f(x) = mx + b \] where \(m\) and \(b\) are constants.
The variable \(x\) is raised to the first power, which makes the graph a straight line.
For example, in the function \(f(x) = 3x + 1\), the number 3 is the slope (\(m\)), and 1 is the y-intercept (\(b\)).

Properties of linear functions:
  • The graph is always a straight line.
  • The domain is all real numbers, because you can plug any real number into \(x\).
  • The slope (\(m\)) tells you how steep the line is.
  • The y-intercept (\(b\)) is the point where the line crosses the y-axis.

Linear functions are common in real-life scenarios. For instance, if you're calculating the total cost with a constant price per item, you are using a linear function.
real numbers
Real numbers are values that can represent a distance along a line.
This includes all the numbers you can think of:
  • Whole numbers like 1, 2, 3
  • Integers like -1, -2, 0
  • Fractions like \(\frac{1}{2}\), \(\frac{3}{4}\)
  • Decimals like 0.5, 2.75
  • Numbers like \(\pi\) and \(\sqrt{2}\)

All these numbers are part of the 'real numbers'.
The domain of many functions includes all real numbers because they accept any real number as an input.
In the example \(f(x) = 3x + 1\), you can substitute any real number for \(x\) and still get a valid output.
This is why the domain of a linear function is written as all real numbers.
interval notation
Interval notation is a way of writing subsets of real numbers and is used to represent the domain and range of functions.
It's a concise way to indicate the span of numbers.
There are different types of intervals you might see:
  • \((a, b)\) means all numbers between \(a\) and \(b\), not including \(a\) and \(b\).
  • \([a, b]\) means all numbers between \(a\) and \(b\), including \(a\) and \(b\).
  • \((a, b]\) or \([a, b)\) are 'half-open' intervals.
  • \((-\fty, a)\), \((a, +\fty)\), \((-\fty, +\fty)\) to indicate unbounded intervals.

For the function \(f(x) = 3x + 1\), the domain is all real numbers.
In interval notation, this is written as \((-\fty, +\fty)\).
This means \(x\) can be any real number from negative infinity to positive infinity without restriction.

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

Find the domain of the function \(f\) given by each of the following. $$f(x)=\frac{3}{2 x^{3}-2 x^{2}-12 x}$$

Perform the indicated operations. $$ \left(9 x^{2}+1\right)-\left(x^{2}+7\right)+\left(4 x^{2}-3 x\right) $$

During the first 13 sec of a jump, a skydiver falls approximately \(11.12 t^{2}\) feet in \(t\) seconds. A small heavy object (with less wind resistance) falls about \(15.4 t^{2}\) feet in \(t\) seconds. Suppose that a skydiver jumps from \(30,000 \mathrm{ft},\) and \(1 \mathrm{sec}\) later a camera falls out of the airplane. How long will it take the camera to catch up to the skydiver?

Simplify. $$ |1.3-(-2.48)| $$

Family has factored a polynomial as \((a-b)(x-y)\) while Jorge has factored the same polynomial as \((b-a)(y-x) .\) Can they both be correct? Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.