SOLUTION: what is the domain of: {{{ sqrt( 15x/(x^2-1) ) }}}

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: what is the domain of: {{{ sqrt( 15x/(x^2-1) ) }}}      Log On


   

Question 975009: what is the domain of: +sqrt%28+15x%2F%28x%5E2-1%29+%29+
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
+sqrt%28+15x%2F%28x%5E2-1%29+%29+

denominator cannot be equal to zero, so exclude all values of x that make denominator equal to zero and they are:
x%5E2-1=0=>x%5E2=1=>x=1 or x=-1 (asymptote)
the domain is:
{ x element R : -1%3Cx%3C=0 or x%3E1}
(assuming a function from reals to reals)

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!

The domain 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%2F%28x%5E2-1%29 >= 0.

The numbers  x+=+1  and  x+=+-1  are not in the domain because the function is not defined for these values.

Further,  the expression  x%2F%28x%5E2-1%29  is positive if and only if the numerator and denominator are both positive at the same time   OR   are both negative at the same time.

The numerator and denominator are both positive at the same time if the following inequalities are held:  x >= 0   AND   abs%28x%29 > 1.
    These two inequalities are valid in the domain  x > 1.

The numerator and denominator are both negative at the same time if the following inequalities are held:  x <= 0   AND   abs%28x%29 < 1.
    These two inequalities are valid in the domain  -1 < x <= 0.

Answer.  The function  sqrt%28x%2F%28x%5E2-1%29%29  has the domain  -1 < x <= 0   and    x > 1   (the union of two sub-domains).