/*! 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 4 Find the domain of each function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 4

Find the domain of each function. $$ g(x)-\frac{2}{x+5} $$

Short Answer

Expert verified
The domain of \( g(x) = \frac{2}{x+5} \) is \( (-\infty, -5) \cup (-5, \infty) \).

Step by step solution

01

Identify the denominator

The denominator of the function \( g(x) = \frac{2}{x+5} \) is \( x+5 \). Set this equal to zero to find the x-value that would make the function undefined.
02

Solve the equation

Solving the equation \( x+5 = 0 \) gives \( x = -5 \). This value would make the denominator of the function equal to zero, so it must be excluded from the domain.
03

Write the domain

The domain of the function \( g(x) = \frac{2}{x+5} \) is all real numbers except \( x = -5 \). In interval notation, this is written as \( (-\infty, -5) \cup (-5, \infty) \).

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.

Function notation
In mathematics, function notation is a convenient and standardized way to write functions. It uses a symbol, most commonly \( f(x)\), to specify a particular function and its variable. The notation helps in expressing the input-output relationship in a clear manner.

Consider a function \(f(x)\). Here, \(f\) represents the name of the function, and \(x\) is the input variable. The entire expression \(f(x)\) stands for the value of the function when \(x\) is the input.

Function notation is crucial because it:
  • Simplifies the process of evaluating expressions at different input values.
  • Makes it easier to read and understand complex functions.
  • Helps in performing operations such as addition, subtraction, or composition of functions.
This notation is universal and applies across different branches of mathematics, making it an essential part of learning and understanding mathematics.
Rational functions
Rational functions are a type of function expressed as the ratio of two polynomials. That is, a rational function \(R(x)\) has the form \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\).

A well-known aspect of rational functions is their defined domain. The function becomes undefined where its denominator is zero. Therefore, finding the domain involves solving \(Q(x) = 0\) and excluding these values.

For rational functions:
  • They can exhibit different types of asymptotes, which are lines that the graph of the function approaches but never touches.
  • Vertical asymptotes occur where the denominator is zero. Horizontal or oblique asymptotes depend on the degrees of the polynomials.
  • They often have interesting behavior near the places where they are undefined, such as vertical asymptotes.
Understanding how rational functions work is vital for analyzing and interpreting their graphical representations and real-world applications.
Interval notation
Interval notation is a system used in mathematics to describe sets of numbers, particularly the domain or range of a function.
This notation employs round or square brackets to represent open or closed intervals.

When using interval notation:
  • Round brackets \((a, b)\) indicate that the endpoints \(a\) and \(b\) are not included in the set (open interval).
  • Square brackets \([a, b]\) indicate that the endpoints \(a\) and \(b\) are included (closed interval).
  • Sometimes, a combination of round and square brackets, such as \((a, b]\) or \([a, b)\), indicates that only one endpoint is included.
  • To represent infinity, such as \((-\infty, b)\) or \((a, \infty)\), always use round brackets since infinity is not a real number and cannot be an endpoint.
This tool is especially useful for expressing the domain of functions where certain values are excluded due to restrictions, such as those found in rational functions.

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

Suppose that \(h(x)=\frac{f(x)}{g(x)} .\) The function \(f\) can be even, odd, or neither. The same is true for the function \(g .\) a. Under what conditions is \(h\) definitely an even function? b. Under what conditions is \(h\) definitely an odd function?

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ r(x)-\frac{1}{2} \sqrt[3]{x-2}+2 $$

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt[3]{-x-2} $$

Suppose that a function \(f\) whose graph contains no breaks or gaps on \((a, c)\) is increasing on \((a, b),\) decreasing on \((b, c)\) and defined at \(b\). Describe what occurs at \(x-b\). What does the function value \(f(b)\) represent?

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=x^{2}-1 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

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.