/*! 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 23 Describe the domain of the funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 23

Describe the domain of the function. $$g(x)=\frac{1}{\sqrt{3-x}}$$

Short Answer

Expert verified
The domain of g(x) is (-鈭�, 3).

Step by step solution

01

Identify the Function's Denominator

The function given is g(x) = \frac{1}{\sqrt{3-x}}.First, focus on the denominator of the function, which is \( \sqrt{3-x} \).
02

Determine Domain Restrictions from the Square Root

For the square root function to be defined, the expression inside the square root, 3 - x, must be greater than 0, because the square root of a negative number is not a real number. Therefore, set up the inequality: 3 - x > 0.
03

Solve the Inequality

Solve the inequality for x: 3 - x > 0 Subtract 3 from both sides: -x > -3 Multiply both sides by -1 (which reverses the inequality sign): x < 3.
04

Write the Domain in Interval Notation

The inequality x < 3 means that x can be any real number less than 3. In interval notation, the domain of the function is (-鈭�, 3).

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.

Inequality
Inequalities play a key role in understanding the domain of a function, especially those involving square roots. For our example, the function g(x) = \frac{1}{\sqrt{3-x}} involves a square root in the denominator. To determine where this function is defined, we need to find where the expression under the square root, 3 - x, remains positive. This situation is described by an inequality: 3 - x > 0.

Key Steps:
  • Set up the inequality based on the rule that square roots of negative numbers are not real: 3 - x > 0.
  • Solve the inequality. In this case:
    \[ 3 - x > 0 \]
    Solving for x gives: \[ x < 3 \]

Understanding how to set up and solve these inequalities ensures that we correctly find where our function is defined.
Interval Notation
Interval notation is a concise way to express the set of numbers that are solutions to the inequality. Once you've solved the inequality, the next step is to write the domain in interval notation.

Steps to Convert Inequality to Interval Notation:
  • From the inequality x < 3, this tells us that x can be any number less than 3.
  • In interval notation, we write this as:
    \[ (-\infty, 3) \]

This notation uses a parenthesis '(', indicating that 3 is not included in the domain. This is important because the square root of zero (which would occur if x = 3) makes the denominator zero, which is undefined.
Square Root Function
Square root functions come with specific rules regarding their domain. For the function g(x) = \frac{1}{\sqrt{3 - x}}, the presence of the square root restricts the domain significantly.

Key Considerations:
  • The expression inside the square root (called the radicand) must be positive.
  • This ensures the square root produces a real number.
  • Hence, we derive the inequality:
    \[ 3 - x > 0 \]

Why is it Important:
  • A square root function is only defined for non-negative numbers.
  • This restricts the domain of functions involving square roots to where the radicand is non-negative.

By understanding this, you can analyze and find the valid input values (domain) of any function involving square roots accurately.

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

A ball thrown straight up into the air has height \(-16 x^{2}+80 x\) feet after \(x\) seconds. (a) Graph the function in the window $$[0,6] \text { by }[-30,120]$$ (b) What is the height of the ball after 3 seconds? (c) At what times will the height be 64 feet? (d) At what time will the ball hit the ground? (e) When will the ball reach its greatest height? What is that height?

Evaluate \(f(4)\). $$f(x)=x^{3}$$

Evaluate \(f(4)\). $$f(x)=x^{-1 / 2}$$

Table 1 shows a conversion table for men's hat sizes for three countries. The function \(g(x)=8 x+1\) converts from British sizes to French sizes, and the function \(f(x)=\frac{1}{8} x\) converts from French sizes to U.S. sizes. Determine the function \(h(x)=f(g(x))\) and give its interpretation. $$\text { Table 1 Conversion Table for Men's Hat Sizes} $$ $$\begin{array}{lcccccccc} \hline \text { Britain } & 6 \frac{1}{2} & 6 \frac{5}{8} & 6 \frac{3}{4} & 6 \frac{7}{8} & 7 & 7 \frac{1}{8} & 7 \frac{1}{4} & 7 \frac{3}{8} \\ \text { France } & 53 & 54 & 55 & 56 & 57 & 58 & 59 & 60 \\ \text { U.S. } & 6 \frac{5}{8} & 6 \frac{3}{4} & 6 \frac{7}{8} & 7 & 7 \frac{1}{8} & 7 \frac{1}{4} & 7 \frac{3}{8} & 7 \frac{1}{2} \\ \hline \end{array}$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$(x y)^{6}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

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.