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Chapter 2: Problem 65

Find the domain of the expression. Use a graphing utility to verify your result. $$\sqrt{\frac{x}{x^{2}-2 x-35}}$$

Short Answer

Expert verified
The domain of the given function is (-Infinity, -5) U (7, Infinity).

Step by step solution

01

Find the roots of the denominator

First, resolve for x in the equation \(x^{2}-2 x-35=0\) using the quadratic formula \( \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), where a=1, b=-2 and c=-35. This gives roots x = 7 and x = -5. These are the values for x that make the denominator zero, therefore, should be excluded from the domain.
02

Determine intervals

Having found the roots, divide the number line into intervals using these roots as boundaries. In this case, intervals would be (-Infinity, -5), (-5, 7) and (7, Infinity).
03

Testing intervals

Take a test point from each interval, and substitute it into the expression inside the square root. If the result is positive, then that interval belongs to the domain. For interval (-Infinity, -5), choose x=-6, for (-5, 7) let x=0 and for (7, Infinity), let x=8.
04

Check the result using a Graphing Utility

Now verify your result using graphing utility such as a graphing calculator or software. If you graph the equation, you will see that it is defined only for the values you have found.

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