SOLUTION: Find the domain and range of the following function in interval form. {{{ sqrt( x^2+3x-4 ) }}} divided by {{{ sqrt( (x-2)^2 ) }}}

Algebra ->  Functions -> SOLUTION: Find the domain and range of the following function in interval form. {{{ sqrt( x^2+3x-4 ) }}} divided by {{{ sqrt( (x-2)^2 ) }}}      Log On


   

Question 1003177: Find the domain and range of the following function in interval form.
+sqrt%28+x%5E2%2B3x-4+%29+ divided by +sqrt%28+%28x-2%29%5E2+%29+
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=sqrt%28x%5E2%2B3x-4%29%2Fsqrt%28+%28x-2%29%5E2+%29+
The restrictions to the domain are
denominators cannot be zero, so sqrt%28+%28x-2%29%5E2+%29%3C%3E0<-->x%3C%3E2 , and
expressions inside square roots cannot be negative, so x%5E2%2B3x-4%3E=0 .
(We know that %28x-2%29%5E2 is not negative).
x%5E2%2B3x-4=%28x-1%29%28x%2B4%29 is negative for -4%3Cx%3C1 ,
so ((-4,1) is not part of the domain,
and we knew that x=2 is not part of the domain, either.
For any other value of x , f%28x%29 is defined, so
the domain of f%28x%29 is %22%28%22-infinity%22%2C+-4+%29+U+%5B+1+%2C+2+%29+U+%28+2+%2C%22infinity%22+%29%22 .

The function cannot be negative, because square roots are non-negative in their domain.
We know f%28x%29=0 for system%28x=-4%2C%22and%22%2Cx=1%29 .
lim%28x-%3E2+%2C+sqrt%28x%5E2%2B3x-4%29%2Fsqrt%28%28x-2%29%5E2%29+%29=lim%28x-%3E2%2Csqrt%282%5E2%2B3%2A2-4%29%2Fsqrt%28%28x-2%29%5E2%29+%29=lim%28x-%3E2%2Csqrt%284%2B6-4%29%2Fsqrt%28%28x-2%29%5E2%29+%29=lim%28x-%3E2%2Csqrt%286%29%2Fsqrt%28%28x-2%29%5E2%29+%29%22=%22%22%2B%22infinity
Since f%28x%29 is continuous in %22%5B+1+%2C+2+%29%22 ,
f%281%29=0 , and lim%28x-%3E2+%2C+f%28x%29+%29%22=%22%22%2B%22infinity ,
f%28x%29 takes all values from 0 to %22%2B%22infinity in %22%5B+1+%2C+2+%29%22 .
The range of f%28x%29 in the interval %22%5B+1+%2C+2+%29%22 is %22%5B+0+%2C%22%22%2B%22infinity%22+%29%22 .
Whatever it does in the rest of its domain, it cannot be negative,
so the range of f%28x%29 across all of its domain is %22%5B+0+%2C%22%22%2B%22infinity%22+%29%22 .