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Chapter 2: Problem 26

Find the domain of each function. $$ h(x)-\sqrt{x-3}+\sqrt{x+4} $$

Short Answer

Expert verified
The domain of the function \( h(x) = \sqrt{x-3} + \sqrt{x+4} \) is \(x \geq 3\).

Step by step solution

01

Solving for \(x - 3\) to be greater than or equal to zero

Firstly, you need to solve for the x-values that will make the expression \(x-3\) greater than or equal to zero. The reason is that the square root of a number, which is less than zero, is not a real number. Solving the inequality \(x-3 \geq 0\), you get \(x \geq 3\). So, for the expression \(x - 3\) under the square root, x should be equal to or greater than 3 for the function to be real.
02

Solving for \(x + 4\) to be greater than or equal to zero

Similarly, for the square root of \(x + 4\) to yield a real number, \(x + 4\) should be greater than or equal to zero. Solving the inequality \(x + 4 \geq 0\), you get \(x \geq -4\). So, for the expression \(x + 4\) under the square root, x should be equal to or greater than -4 for the function to be real.
03

Determining the domain

Having solved for x in both inequalities, you can determine the domain of the function. The domain of the function will be the intersection of the solutions from both inequalities since x must satisfy both conditions for the function to be defined. The intersection of \(x \geq 3\) and \(x \geq -4\) is \(x \geq 3\). Therefore, the domain of the function \(h(x) = \sqrt{x-3} + \sqrt{x+4}\) is \(x \geq 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
The square root function, often represented as \( \sqrt{x} \), is a function that returns the non-negative square root of a number \( x \). For example, \( \sqrt{9} = 3 \), because 3 multiplied by itself equals 9.
  • The square root function is only defined for non-negative numbers \( x \geq 0 \). This is because the square root of a negative number is not a real number.
  • In the context of our problem, we have two expressions under the square roots: \( \sqrt{x-3} \) and \( \sqrt{x+4} \).
For \( \sqrt{x-3} \) to be defined, \( x-3 \) must be greater than or equal to zero. Similarly, for \( \sqrt{x+4} \), \( x+4 \) must be greater than or equal to zero. Understanding these conditions is key to finding the domain of any function involving square roots.
Inequalities
Inequalities are mathematical expressions used to compare two values, often involving symbols such as \( >, <, \geq, \) or \( \leq \). In this exercise, inequalities help us find conditions under which the square root function is defined.
  • The inequality \( x - 3 \geq 0 \) simplifies to \( x \geq 3 \). This tells us that \( x \) must be 3 or greater for the square root expression \( \sqrt{x-3} \) to be a real number.
  • The inequality \( x + 4 \geq 0 \) simplifies to \( x \geq -4 \). This indicates that \( x \) must be -4 or greater for \( \sqrt{x+4} \) to be real, allowing values like \( x = -4, -3, \text{etc.} \)
Inequalities help define the range of permissible \( x \) values for each square root expression. When considering the domain of the entire function, we focus on the overlap where both inequalities are satisfied.
Real Numbers
Real numbers include all the numbers on the number line. These cover both rational numbers, like fractions and integers, and irrational numbers, like \( \sqrt{2} \) or \( \pi \).
  • Real numbers are infinite and provide a complete set of values for most algebraic operations, including addition and subtraction, multiplication and division, as well as square roots.
  • In our problem, the terms under the square roots need to be real numbers for \( h(x) = \sqrt{x-3} + \sqrt{x+4} \) to be defined.
The domain of a function is a set of real numbers for which the function is defined. In our problem, by combining conditions from inequalities, the domain is defined for \( x \geq 3 \). This ensures that both square roots yield real numbers, thereby making the entire expression valid within the set of real numbers.

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