/*! 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 94 Determine the domain of each fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 94

Determine the domain of each function described. $$ g(x)=\sqrt{x+8} $$

Short Answer

Expert verified
Domain: \( [-8, \infty) \)

Step by step solution

01

Understand the function

The given function is \( g(x) = \sqrt{x + 8} \). This is a square root function, and the expression inside the square root must be non-negative for the function to be defined.
02

Set up the inequality

To determine the domain, set the expression inside the square root greater than or equal to zero: \( x + 8 \geq 0 \).
03

Solve the inequality

Solve for \( x \) in the inequality: \( x + 8 \geq 0 \) Subtract 8 from both sides to get: \( x \geq -8 \).
04

Write the domain in interval notation

The solution to the inequality \( x \geq -8 \) indicates that \( x \) can take any value starting from \( -8 \) and going to infinity. In interval notation, this is written as \( [-8, \infty) \).

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.

Square Root Function
A square root function includes the square root (\( \sqrt{ } \)) of a variable or an expression. The simplest form is \( \sqrt{x} \). One key property is that the expression inside the square root, called the radicand, must be non-negative. This is because the square root of a negative number is not defined in the set of real numbers. In our exercise, the function is \( g(x) = \sqrt{x + 8} \). Here, \( x + 8 \) is the radicand and must be non-negative to keep the function defined for real numbers.
Inequality
Inequalities help determine the range of values that a variable can take. In this case, we need to ensure the radicand (\( x + 8 \)) is non-negative. We set it up as the inequality \( x + 8 \geq 0 \). Solving this inequality, we subtract 8 from both sides, resulting in \( x \geq -8 \).
This tells us that \( x \) must be at least \(-8\). Any smaller value for \( x \) would make the radicand negative, rendering the square root undefined. To recap:
- Setup the inequality based on the square root function.
- Solve for the variable while ensuring the radicand is non-negative.
Interval Notation
Interval notation provides a concise way to describe the set of numbers that form the domain of the function. After solving the inequality \( x \geq -8 \), we need to express this range properly. The solution \( x \geq -8 \) tells us \( x \) can be any number greater than or equal to \( -8 \) and extends to infinity. In interval notation, this is written as:
- The starting value is \( -8 \), included in the set, represented by a square bracket [ .
- The set extends to infinity, represented by the symbol \( \infty \), with a parenthesis ( because infinity is not a real number and hence not included.
Together, the domain of \( g(x) = \sqrt{x + 8} \) is: \( [-8, \infty) \).

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

Simplify. $$ 7 x \sqrt{(x+y)^{3}}-5 x y \sqrt{x+y}-2 y \sqrt{(x+y)^{3}} $$

f(x)\( and \)g(x)\( are as given. Find \)(f \cdot g)(x) \cdot$ Assume that all variables represent non-negative real numbers. $$ f(x)=x+\sqrt{7}, g(x)=x-\sqrt{7} $$

Describe a procedure that uses the distance formula to determine whether three points, \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right),\) are vertices of a right triangle.

Contracting. Cleary Construction has a hose attached to their insulation blower that permits them to reach anywhere in a circular area of \(6160 \mathrm{ft}^{2} .\) Find the dimensions of the largest square room with 12 -ft ceilings in which they could reach all corners with the hose while leaving the blower centrally located. Assume that the blower sits on the floor.

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt{a^{4} b^{3} c^{4}} \sqrt[3]{a b^{2} c} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.