/*! 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 22 Determine whether the statement ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 22

Determine whether the statement is true or false. Explain your answer. If \(g(x)=1 / \sqrt{f(x)},\) then the domain of \(g\) consists of all those real numbers \(x\) for which \(f(x) \neq 0\)

Short Answer

Expert verified
True, \( g(x) \) is defined when \( f(x) > 0 \), aligning with \( f(x) \neq 0 \).

Step by step solution

01

Understanding the Domain of a Square Root Function

The function is defined as \( g(x) = \frac{1}{\sqrt{f(x)}} \). For the square root \( \sqrt{f(x)} \) to be a real number, \( f(x) \) must be greater than zero, since the square root of a negative number is not a real number. Thus, we require \( f(x) > 0 \).
02

Considering the Denominator

Since \( g(x) = \frac{1}{\sqrt{f(x)}} \), the expression is undefined for \( \sqrt{f(x)} = 0 \), because division by zero is undefined. This means \( \sqrt{f(x)} eq 0 \), leading to \( f(x) eq 0 \).
03

Combining Both Conditions

From both steps, the condition \( f(x) eq 0 \) covers both requirements of \( f(x) > 0 \) (since \( f(x) \) must be positive) and \( \sqrt{f(x)} eq 0 \). This means \( f(x) > 0 \), allowing for the numbers that make the function defined and real.
04

Conclusion

Hence, the domain of \( g(x) \) is indeed all those real numbers \( x \) for which \( f(x) > 0 \). The original statement suggests \( f(x) eq 0 \), which agrees with \( f(x) > 0 \). Therefore, the statement aligns with our conclusion.

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.

Square Root Function
The square root function is key when determining the domain of certain mathematical functions. Specifically, in the expression \( g(x) = \frac{1}{\sqrt{f(x)}} \), the square root \( \sqrt{f(x)} \) needs to be a real number. Key insights include:
  • The square root of non-negative numbers (i.e., numbers greater than or equal to zero) yields a real result. However, the square root of a negative number is not a real number in the standard real number system.
  • This requirement implies that for the expression \( \sqrt{f(x)} \) to be valid and yield a real number, \( f(x) \) must be positive.
Thus, to ensure the expression is defined, \( f(x) > 0 \) is a necessary condition.
Numerical Inequality
Numerical inequalities play an essential role in identifying the domain of functions with constraints. In the problem, the inequality \( f(x) > 0 \) defines the valid range for \( x \):
  • This inequality shows that \( f(x) \) needs to be strictly greater than zero, ensuring that the function \( g(x) \) remains valid.
  • Inequalities are useful for keeping domains limited to values that preserve real number solutions, which means avoiding undefined or complex numbers in the context of square roots and denominators.
By enforcing \( f(x) > 0 \), we ensure \( g(x) \) is consistently defined over its entire real number domain.
Division by Zero
Division by zero is undefined in mathematics, which impacts the domain of functions like \( g(x) = \frac{1}{\sqrt{f(x)}} \). When discussing this concept:
  • It is crucial that \( \sqrt{f(x)} eq 0 \) so that the denominator remains non-zero, thereby avoiding an undefined expression.
  • Whenever the denominator of a fraction equates to zero, the function becomes undefined as dividing by zero does not yield a finite or meaningful result.
Therefore, ensuring \( f(x) eq 0 \) is necessary, and this circumstance aligns with the more stringent condition \( f(x) > 0 \).
Real Numbers
Real numbers encompass a vast set, including all rational and irrational numbers. For both the square root function and division considerations, understanding real numbers is vital:
  • Real numbers are those that can be found on the number line, excluding imaginary numbers, such as those derived from negative square roots.
  • Any functions or expressions operating within real numbers must adhere to conditions that do not elicit complex results, such as taking the square root of a negative number or performing division by zero.
For \( g(x) \), this means the domain consists of real values \( x \) such that \( f(x) > 0 \), ensuring the entire expression stays within the realm of 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 graph of the equation by making appropriate transformations to the graph of a basic power function. If you have a graphing utility, use it to check your work. $$ \begin{array}{ll}{\text { (a) } y=1-\sqrt{x+2}} & {\text { (b) } y=1-\sqrt[3]{x+2}} \\ {\text { (c) } y=\frac{5}{(1-x)^{3}}} & {\text { (d) } y=\frac{2}{(4+x)^{4}}}\end{array} $$

Find the amplitude and period, and sketch at least two periods of the graph by hand. If you have a graphing utility, use it to check your work. $$ \begin{array}{ll}{\text { (a) } y=-1-4 \sin 2 x} & {\text { (b) } y=\frac{1}{2} \cos (3 x-\pi)} \\ {\text { (c) } y=-4 \sin \left(\frac{x}{3}+2 \pi\right)} & {}\end{array} $$

Find a formula for \(f^{-1}(x)\) $$ f(x)=\left\\{\begin{array}{ll}{2 x,} & {x \leq 0} \\ {x^{2},} & {x>0}\end{array}\right. $$

Determine whether the statement is true or false. Explain your answer. The range of the inverse tangent function is the interval \(-\pi / 2 \leq y \leq \pi / 2\).

A variable \(y\) is said to be inversely proportional to the square of a variable \(x\) if \(y\) is related to \(x\) by an equation of the form \(y=k / x^{2},\) where \(k\) is a nonzero constant, called the constant of proportionality. This terminology is used in these exercises. It follows from Newton's Law of Universal Gravitation that the weight \(W\) of an object (relative to the Earth) is inversely proportional to the square of the distance \(x\) between the object and the center of the Earth, that is, \(W=C / x^{2}\). (a) Assuming that a weather satellite weighs 2000 pounds on the surface of the Earth and that the Earth is a sphere of radius 4000 miles, find the constant \(C\). (b) Find the weight of the satellite when it is 1000 miles above the surface of the satellite when it is 1000 miles (c) Make a graph of the satellite's weight versus its distance from the center of the Earth. (d) Is there any distance from the center of the Earth at which the weight of the satellite is zero? Explain your reasoning.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.