/*! 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 1 Determine the domain of the func... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 1

Determine the domain of the function. $$f(x)=\frac{x^{2}}{2-x}$$

Short Answer

Expert verified
The domain is all real numbers except \(x = 2\), or \[(-\infty, 2) \cup (2, \infty)\]

Step by step solution

01

Understand the Function

Consider the given function: \[ f(x) = \frac{x^2}{2 - x} \]The domain of a function includes all possible values of \(x\) for which the function is defined.
02

Identify Restrictions in the Denominator

The function is undefined when the denominator is zero. Set the denominator equal to zero and solve for \(x\):\[ 2 - x = 0 \]Solving this equation gives:\[ x = 2 \]
03

Establish the Domain

Since the function is undefined at \(x = 2\), it must be excluded from the domain. The domain is all real numbers except \(x = 2\). In interval notation, the domain is:\[ (-\infty, 2) \cup (2, \infty) \]

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.

precalculus
Precalculus is a course that lays the foundation for calculus. It involves the study of functions, algebra, and trigonometry. Understanding these concepts is crucial before moving on to more advanced mathematical topics. One of the key skills in precalculus is determining the domain of a function. The domain is all the possible input values (usually represented as \(x\)) for which the function is defined.
interval notation
Interval notation is a way to describe a set of numbers along a number line. It's particularly useful in specifying the domain and range of functions. In interval notation, we use brackets and parentheses to denote subsets of real numbers:
  • Brackets [ ] indicate that the endpoint is included in the interval (inclusive).
  • Parentheses ( ) indicate that the endpoint is not included (exclusive).
For example, if we want to describe all real numbers except \(x = 2\), we use \((-\infty, 2) \cup (2, \infty)\). This tells us that the function is valid for all values of \(x\) except 2.
undefined function values
When working with functions, it's important to recognize when they become undefined. A function is undefined if any of its operations lead to something that doesn't make sense mathematically, such as division by zero. In our example, the function \( f(x) = \frac{x^{2}}{2 - x} \) becomes undefined when the denominator is zero:
  • We set the denominator equal to zero: \( 2 - x = 0 \).
  • Solving for \( x \) gives \( x = 2 \).
This tells us that the function is unable to take the value of 2, which must be excluded from the domain.
The domain, therefore, is all real numbers except \( x = 2 \). In interval notation, it's written as \((-\infty, 2) \cup (2, \infty)\). This indicates the function's values span all real numbers except where \( x = 2 \).

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

List the critical values of the related function. Then solve the inequality. $$4 \geq \frac{4}{x}+x$$

List the critical values of the related function. Then solve the inequality. $$\frac{2}{x^{2}+3} > \frac{3}{5+4 x^{2}}$$

Find the nonlinear asymptote of the function. $$f(x)=\frac{x^{5}+2 x^{3}+4 x^{2}}{x^{2}+2}$$

Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible. Vertical asymptotes \(x=-4, x=5 ;\) horizontal asymptote \(y=\frac{3}{2} ; x\) -intercept \((-2,0)\)

The Hold-It Container Co. is designing an open-top rectangular box, with a square base, that will hold 108 cubic centimeters. (Image can't copy) a) Express the surface area \(S\) as a function of the length \(x\) of a side of the base. b) Use a graphing calculator to graph the function on the interval \((0, \infty)\) c) Estimate the minimum surface area and the value of \(x\) that will yield it.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.