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Chapter 1: Problem 18

Find the domain of each function. $$f(x)=\sqrt{x+2}$$

Short Answer

Expert verified
The domain of the function \(f(x)=\sqrt{x+2}\) is \(x \geq -2\).

Step by step solution

01

Determining Inequality

To find the domain, we set the expression under the square root greater than or equal to zero. So, \(x+2 \geq 0\)
02

Solving for x

Now, we isolate x in the inequality. Subtract 2 from both sides to isolate x, we get: \(x \geq -2\).
03

Determining Domain

The solution to this inequality is the domain of the function. The domain of function \(f(x)=\sqrt{x+2}\) is \(x \geq -2\).

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Key Concepts

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

Understanding Square Root Functions
The square root function is a type of function that involves the expression \(\sqrt{x}\). It is commonly recognized by its unique shape and behavior in graphs. Essentially, the square root function can be thought of as the inverse of squaring a number. When working with square roots, it is important to remember:
  • Square roots are defined only for non-negative values under the square root. This is because the square root of a negative number results in an imaginary number, which is not typically represented on a real number line.
  • The smallest value under a square root for real numbers is zero.
In practical terms, this means if you have a function like \(f(x) = \sqrt{x+2}\), you need to find what values of \(x\) make the expression inside the square root non-negative. This understanding helps greatly when we move on to finding the domain of such functions.
Demystifying Inequalities
Inequalities are fundamental in determining the domains of functions, especially those involving square roots. They allow you to express a range of possible solutions rather than a single answer. Here are some key aspects:
  • An inequality states that one side of the equation is either greater than or less than the other side.
  • For square root functions, you typically set the expression under the square root to be greater than or equal to zero.
Consider the inequality \(x + 2 \geq 0\). It tells us \(x\) values should make this expression non-negative. Solving such inequalities involves certain steps to isolate \(x\), giving insight into which \(x\) values satisfy the original condition. Understanding inequalities is crucial for defining function domains as it ensures the expression under the root is appropriate for real number solutions.
Effectively Solving Inequalities
Solving inequalities is about finding the range of \(x\) values that make the inequality true. Let's consider the inequality \(x + 2 \geq 0\):
  • You start by positioning the inequality such that \(x\) is isolated. For \(x + 2 \geq 0\), subtract 2 from both sides, resulting in \(x \geq -2\).
  • The solution, \(x \geq -2\), tells us the domain of the function \(f(x) = \sqrt{x+2}\), meaning any \(x\) equal to or greater than \(-2\) is permissible.
Solving inequalities often involves simple algebraic manipulations like addition or subtraction, as seen here, to form easy-to-understand constraints. These steps ensure you correctly define the function's domain, aligning with the constraints dictated by the nature of the square root.

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Most popular questions from this chapter

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x),\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\) and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(c>1\) e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

Find the area of the donut-shaped region bounded by the graphs of \((x-2)^{2}+(y+3)^{2}=25\) and \((x-2)^{2}+(y+3)^{2}=36\)

Graph \(y_{1}=x^{2}-2 x, y_{2}=x,\) and \(y_{3}=y_{1} \div y_{2}\) in the same \([-10,10,1]\) by \([-10,10,1]\) viewing rectangle. Then use the TRACE feature to trace along \(y_{3}\). What happens at \(x=0 ?\) Explain why this occurs.

Exercises \(101-103\) will help you prepare for the material covered in the next section. Write an algebraic expression for the fare increase if a \(\$ 200\) plane ticket is increased to \(x\) dollars.

$$\text { Solve for } y: \quad A x+B y=C y+D$$
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