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Chapter 7: Problem 19

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{5}{x^{2}+1}$$

Short Answer

Expert verified
The given expression \(\frac{5}{x^{2}+1}\) is defined for all real numbers.

Step by step solution

01

Identify the Denominator

The denominator of the given expression is \(x^{2}+1\). It is the part of the expression which, if equals to zero, will make the expression undefined.
02

Solve the Denominator Equals to Zero

In order to make sure when the expression will be undefined, solve the equation \(x^{2}+1 = 0\).
03

Result and Discussion

The solution of the equation \(x^{2}+1 = 0\) results in complex numbers, this happens because when you subtract 1 from both sides you get \(x^{2} = -1\) and taking square root of -1 results in an imaginary number. This means the expression is never undefined for any real number \(x\).

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Most popular questions from this chapter

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}+4 x+3}{x+2}-\frac{5 x+9}{x+2}=x-2, x \neq-2$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{x-1}+\frac{3}{x+2}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{x}+9$$

Two skiers begin skiing along a trail at the same time. The faster skier averages 9 miles per hour and the slower skier averages 6 miles per hour. The faster skier completes the trail \(\frac{1}{4}\) hour before the slower skier. How long is the trail?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.
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