/*! 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 62 Find the domain of the expressio... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 62

Find the domain of the expression. Use a graphing utility to verify your result. $$\sqrt{x^{2}-4}$$

Short Answer

Expert verified
The domain of the function \(\sqrt{x^{2}-4}\) is \((- \infty, -2] \cup [2, \infty)\)

Step by step solution

01

Identify the Expression Inside the Square Root

Looking at the equation we can identifythe expression as \(x^{2}-4\). This is the expression inside the square root.
02

Set the Expression Greater Than or Equal to Zero

To find the valid domain for this function, set the expression inside the square root greater than or equal to zero, i.e. \(x^{2}-4 \geq 0\).
03

Solve for x

To find the values of x that satisfy the inequality, we solve it. The values of x will be any number that is less than or equal to -2 and any number that is greater than or equal to 2. So the solution is \(x \leq -2\) and \(x \geq 2\).
04

Write the Domain in Interval Notation

The final step is to write the domain in interval notation. The solution \(-\infty < x \leq -2\) and \(2 \leq x < \infty\) can be written as \((- \infty, -2] \cup [2, \infty)\)
05

Verify with Graphing Utility

The last step is to verify the solution with a graphing utility. Plot the function \(\sqrt{x^{2}-4}\) on a graphing calculator or software. The graph should reflect the domain as \((- \infty,-2] \cup [2, \infty)\).

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(z)=z^{2}-2 z+2$$

The mean salaries \(S\) (in thousands of dollars) of public school classroom teachers in the United States from 2000 through 2011 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, \(S\) } \\\\\hline 2000 & 42.2 \\\2001 & 43.7 \\\2002 & 43.8 \\\2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\2008 & 51.3 \\\2009 & 52.9 \\\2010 & 54.4 \\\2011 & 54.2 \\\\\hline\end{array}$$ A model that approximates these data is given by $$S=\frac{42.16-0.236 t}{1-0.026 t}, \quad 0 \leq t \leq 11$$ where \(t\) represents the year, with \(t=0\) corresponding to 2000. (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) Use the model to predict when the salary for classroom teachers will exceed \(\$ 60,000\). (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.

Use the position equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ where \(s\) represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 160 feet per second. (a) At what instant will it be back at ground level? (b) When will the height exceed 384 feet?

For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each function. Describe the relationship between the degree of the function and the sign of the leading coefficient of the function and the right-hand and left-hand behavior of the graph of the function. (a) \(f(x)=x^{3}-2 x^{2}-x+1\) (b) \(f(x)=2 x^{5}+2 x^{2}-5 x+1\) (c) \(f(x)=-2 x^{5}-x^{2}+5 x+3\) (d) \(f(x)=-x^{3}+5 x-2\) (e) \(f(x)=2 x^{2}+3 x-4\) (f) \(f(x)=x^{4}-3 x^{2}+2 x-1\) (g) \(f(x)=x^{2}+3 x+2\)

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=2 x^{4}-8 x+3\) (a) Upper: \(x=3\) (b) Lower: \(x=-4\)
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