Question 975009


The <B>domain</B> for this function is the set of real numbers where the expression under the square root is defined and is greater or equal to zero (is non-negative):


{{{x/(x^2-1)}}} >= {{{0}}}.


The numbers &nbsp;{{{x = 1}}}&nbsp; and &nbsp;{{{x = -1}}}&nbsp; are not in the domain because the function is not defined for these values. 


Further, &nbsp;the expression &nbsp;{{{x/(x^2-1)}}}&nbsp; is positive if and only if the numerator and denominator are both positive at the same time &nbsp;&nbsp;OR&nbsp;&nbsp; are both negative at the same time. 


The numerator and denominator are both positive at the same time if the following inequalities are held: &nbsp;{{{x}}} >= {{{0}}} &nbsp;&nbsp;AND&nbsp;&nbsp; {{{abs(x)}}} > 1.

&nbsp;&nbsp;&nbsp;&nbsp;These two inequalities are valid in the domain &nbsp;{{{x}}} > 1. 


The numerator and denominator are both negative at the same time if the following inequalities are held: &nbsp;{{{x}}} <= {{{0}}} &nbsp;&nbsp;AND&nbsp;&nbsp; {{{abs(x)}}} < 1.

&nbsp;&nbsp;&nbsp;&nbsp;These two inequalities are valid in the domain &nbsp;{{{-1}}} < {{{x}}} <= {{{0}}}. 


<B>Answer</B>. &nbsp;The function &nbsp;{{{sqrt(x/(x^2-1))}}}&nbsp; has the domain &nbsp;{{{-1}}} < {{{x}}} <= {{{0}}} &nbsp;&nbsp;and&nbsp;&nbsp; &nbsp;{{{x}}} > 1 &nbsp;&nbsp;(the union of two sub-domains).