/*! 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 27 Specify the domain for each of t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 27

Specify the domain for each of the functions. $$h(x)=\sqrt{x+4}$$

Short Answer

Expert verified
The domain of \( h(x) = \sqrt{x+4} \) is \( [-4, \infty) \).

Step by step solution

01

Identify the Function Type

The given function is \( h(x) = \sqrt{x+4} \). It involves a square root, which means we must ensure that the expression inside the square root is non-negative to find the domain.
02

Set the Expression Inside the Square Root to be Non-negative

For the function \( h(x) \) to be defined, the expression inside the square root must be greater than or equal to zero. So, we set up the inequality: \( x+4 \geq 0 \).
03

Solve the Inequality

Solve the inequality from Step 2 to find the values of \( x \) for which the function is defined. Subtract 4 from both sides: \( x \geq -4 \).
04

Write the Domain in Interval Notation

The solution to \( x \geq -4 \) can be written in interval notation as \( [-4, \infty) \). This tells us that \( x \) must be greater than or equal to -4 for the function to be defined.

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 Functions
Square root functions are a special type of mathematical functions that include a square root symbol. For such functions, the expression contained inside the square root must be non-negative. This ensures that the function outputs real numbers, as the square root of a negative number is not defined within the real number system.
For example, with the function \[ h(x) = \sqrt{x+4} \], we examine the expression inside the square root, which is \( x+4 \). This needs to be zero or positive. This requirement leads us to the concept of inequalities.
Inequalities
Inequalities are equations that utilize symbols like >, <, 鈮�, and 鈮� to show the relationship between expressions or numbers. When dealing with square root functions, inequalities help us determine the domain by stating the necessary conditions for the expression inside the root to be non-negative.
For the function \[ h(x) = \sqrt{x+4} \], we established the inequality \( x+4 \geq 0 \) to ensure the expression is valid for real numbers. Solving this inequality involves simple algebraic manipulation, such as subtracting 4 from both sides, to yield the solution \( x \geq -4 \).
This solution then guides us in expressing the domain using interval notation.
Interval Notation
Interval notation is a method of describing sets of numbers, typically denoting the domain or range of a function. With interval notation, brackets \([ \ ]\) indicate that an endpoint is included in the interval, while parentheses \(( \ )\) signify that the endpoint is not included.
For the solution to \( x \geq -4 \), the interval notation is written as \[ [-4, \infty) \], to indicate that the domain includes all numbers from -4 to infinity, with -4 itself being part of the domain.
  • The square bracket \([ \ ]\) around -4 shows us that -4 is included in the domain.
  • The parenthesis \(( \ )\) around infinity reflects that infinity is not an actual number, hence not included.

Thus, the domain of the function \( h(x) = \sqrt{x+4} \) can be fully described as \([-4, \infty)\).

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

Determine the indicated functional values. (Objective 2 ) If \(f(x)=-x^{3}\) and \(g(x)=|2 x+4|\), find \((f \circ g)(-1)\) and \((g \circ f)(-3)\).

Does the function \(f(x)=4\) have an inverse? Explain your answer.

The volume of a cylinder varies jointly as its altitude and the square of the radius of its base. If a cylinder that has a base with a radius of 5 meters and has an altitude of 7 meters has a volume of \(549.5\) cubic meters, find the volume of a cylinder that has a base with a radius of 9 meters and has an altitude of 14 meters.

The simple interest earned by a certain amount of money varies jointly as the rate of interest and the time (in years) that the money is invested. If the money is invested at \(8 \%\) for 2 years, \(\$ 80\) is earned. How much is earned if the money is invested at \(6 \%\) for 3 years?

Translate each statement of variation into an equation, and use \(k\) as the constant of variation. At a constant temperature, the volume \((V)\) of a gas varies inversely as the pressure \((P)\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.