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Chapter 1: Problem 101

Domain Determine the values of the variable for which the expression is defined as a real number. $$\sqrt{x^{2}-9}$$

Short Answer

Expert verified
The expression is defined for \( x \leq -3 \) and \( x \geq 3 \).

Step by step solution

01

Understand the Expression

The expression we are given is \( \sqrt{x^2 - 9} \). Our goal is to determine for which values of \( x \) this expression is a real number.
02

Identify the Condition for a Real Square Root

A square root expression, \( \sqrt{a} \), is defined as a real number when \( a \geq 0 \). We need to apply this condition to the expression under the square root.
03

Set Up Inequality

Set up the inequality based on the condition for a real square root: \( x^2 - 9 \geq 0 \). This will determine the values of \( x \) that make the expression a real number.
04

Solve the Inequality

We need to solve the inequality \( x^2 - 9 \geq 0 \) to find the values of \( x \). This inequality can be factored as \((x-3)(x+3) \geq 0\).
05

Test Intervals

The critical points from the factors \((x-3)\) and \((x+3)\) are \(x = 3\) and \(x = -3\). Check each interval: \((-\infty, -3)\), \([-3, 3]\), \((3, \infty)\) to see where the inequality holds.
06

Determine Valid Intervals

When testing these intervals, the inequality \((x-3)(x+3) \geq 0\) holds for \(x \leq -3\) and \(x \geq 3\) but not for \((-3, 3)\). Thus, the expression is real for these intervals.
07

Express the Domain

The domain of the expression, where it is defined as a real number, is \(( -\infty, -3] \cup [3, \infty)\).

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Key Concepts

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

Real Number
When dealing with mathematical expressions, it often involves understanding the concept of a real number. Real numbers include all the numbers on the number line. This means both positive and negative numbers, zero, and fractions or decimals. However, it doesn't include imaginary or complex numbers, like the square root of a negative number.
For an expression involving a square root to be considered a real number, the part inside the square root must be non-negative. For instance, in the expression \(\sqrt{x^2 - 9}\), the term \(x^2 - 9\) must be greater than or equal to zero, which is why finding where the expression becomes a real number requires solving related inequalities.
Square Root Inequality
When we have an inequality with a square root, such as \(\sqrt{x^2 - 9}\), the goal is to identify the range of values for \(x\) that keeps the expression inside the root non-negative. This turns into a simple mathematical condition: \(x^2 - 9 \geq 0\).

To solve, rewrite this inequality by recognizing it as a difference of squares: \((x-3)(x+3) \geq 0\). This method allows you to factor and easily identify critical points. These are the points at which the expression changes, occurring at \(x = 3\) and \(x = -3\). Identifying these points is essential to determine the intervals where the inequality holds true.
Interval Testing
Upon identifying critical points \(x = 3\) and \(x = -3\), the task is to test intervals around these points to determine where the inequality holds. These intervals are \((-\infty, -3)\), \([-3, 3]\), and \((3, \infty)\).
  • In \((-\infty, -3)\), select any number, say \(x = -4\). Plugging into \((x-3)(x+3) \geq 0\), it results in a positive value, meaning this interval satisfies the inequality.
  • In \([-3, 3]\), take \(x = 0\). The product \((0-3)(0+3) = -9\), which does not satisfy the condition of being non-negative.
  • In \((3, \infty)\), choose \(x = 4\). Here, \((4-3)(4+3) = 7\), a positive outcome that satisfies the inequality.
Thus, the valid values of \(x\) for the expression to remain real are in the intervals \((-\infty, -3]\) and \([3, \infty)\), excluding the section between \(-3\) and \(3\) where the result turns negative.

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Most popular questions from this chapter

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt[5]{x^{3} y^{2}} \sqrt[19]{x^{4} y^{16}}\) (b) \(\frac{\sqrt[3]{8 x^{2}}}{\sqrt{x}}\)

Prove the following Laws of Exponents for the case in which \(m\) and \(n\) are positive integers and \(m>n\) (a) Law \(2: \frac{a^{m}}{a^{n}}=a^{m-n}\) (b) Law \(5:\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}\)

Depth of a Well One method for determining the depth of a well is to drop a stone into it and then measure the time it takes until the splash is heard. If \(d\) is the depth of the well (in feet) and \(t_{1}\) the time (in seconds) it takes for the stone to fall, then \(d=16 t_{1}^{2},\) so \(t_{1}=\sqrt{d} / 4 .\) Now if \(t_{2}\) is the time it takes for the sound to travel back up, then \(d=1090 t_{2}\) because the speed of sound is 1090 ft's. So \(t_{2}=d / 1090\) Thus the total time elapsed between dropping the stone and hearing the splash is $$t_{1}+t_{2}=\frac{\sqrt{d}}{4}+\frac{d}{1090}$$ How deep is the well if this total time is 3 s? PICTURE CANT COPY

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DISCUSS: Proof That \(0=1 ?\) The following steps appear to give equivalent equations, which seem to prove that \(1=0 .\) Find the error. \(\begin{aligned} x &=1 & & \text { Given } \\ x^{2} &=x & & \text { Multiply by } x \\ x^{2}-x &=0 & & \text { Subtract } x \end{aligned}$$(x-1)=0\)Factor \(\frac{(x-1)}{x-1}=\frac{0}{x-1} \quad\) Divide by \(x-1$$x=0 \quad\) Simplify\(1=0 \quad\) Given \(x=1\)
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