document.write( "Question 1003177: Find the domain and range of the following function in interval form. \r
\n" ); document.write( "\n" ); document.write( " $\"+sqrt%28+x%5E2%2B3x-4+%29+\"$ divided by $\"+sqrt%28+%28x-2%29%5E2+%29+\"$ \n" ); document.write( "

Algebra.Com's Answer #620107 by KMST(5328) $\"\"$ $\"About$

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$\"f%28x%29=sqrt%28x%5E2%2B3x-4%29%2Fsqrt%28+%28x-2%29%5E2+%29+\"$
\n" ); document.write( "The restrictions to the domain are
\n" ); document.write( "denominators cannot be zero, so $\"sqrt%28+%28x-2%29%5E2+%29%3C%3E0\"$ <--> $\"x%3C%3E2\"$ , and
\n" ); document.write( "expressions inside square roots cannot be negative, so $\"x%5E2%2B3x-4%3E=0\"$ .
\n" ); document.write( "(We know that $\"%28x-2%29%5E2\"$ is not negative).
\n" ); document.write( " $\"x%5E2%2B3x-4=%28x-1%29%28x%2B4%29\"$ is negative for $\"-4%3Cx%3C1\"$ ,
\n" ); document.write( "so ((-4,1) is not part of the domain,
\n" ); document.write( "and we knew that $\"x=2\"$ is not part of the domain, either.
\n" ); document.write( "For any other value of $\"x\"$ , $\"f%28x%29\"$ is defined, so
\n" ); document.write( "the domain of $\"f%28x%29\"$ is $\"%22%28%22\"$ $\"-infinity\"$ $\"%22%2C+-4+%29+U+%5B+1+%2C+2+%29+U+%28+2+%2C%22\"$ $\"infinity\"$ $\"%22+%29%22\"$ .
\n" ); document.write( "
\n" ); document.write( "The function cannot be negative, because square roots are non-negative in their domain.
\n" ); document.write( "We know $\"f%28x%29=0\"$ for $\"system%28x=-4%2C%22and%22%2Cx=1%29\"$ .
\n" ); document.write( " $\"lim%28x-%3E2+%2C+sqrt%28x%5E2%2B3x-4%29%2Fsqrt%28%28x-2%29%5E2%29+%29\"$ = $\"lim%28x-%3E2%2Csqrt%282%5E2%2B3%2A2-4%29%2Fsqrt%28%28x-2%29%5E2%29+%29\"$ = $\"lim%28x-%3E2%2Csqrt%284%2B6-4%29%2Fsqrt%28%28x-2%29%5E2%29+%29\"$ = $\"lim%28x-%3E2%2Csqrt%286%29%2Fsqrt%28%28x-2%29%5E2%29+%29\"$ $\"%22=%22\"$ $\"%22%2B%22\"$ $\"infinity\"$
\n" ); document.write( "Since $\"f%28x%29\"$ is continuous in $\"%22%5B+1+%2C+2+%29%22\"$ ,
\n" ); document.write( " $\"f%281%29=0\"$ , and $\"lim%28x-%3E2+%2C+f%28x%29+%29\"$ $\"%22=%22\"$ $\"%22%2B%22\"$ $\"infinity\"$ ,
\n" ); document.write( " $\"f%28x%29\"$ takes all values from $\"0\"$ to $\"%22%2B%22\"$ $\"infinity\"$ in $\"%22%5B+1+%2C+2+%29%22\"$ .
\n" ); document.write( "The range of $\"f%28x%29\"$ in the interval $\"%22%5B+1+%2C+2+%29%22\"$ is $\"%22%5B+0+%2C%22\"$ $\"%22%2B%22\"$ $\"infinity\"$ $\"%22+%29%22\"$ .
\n" ); document.write( "Whatever it does in the rest of its domain, it cannot be negative,
\n" ); document.write( "so the range of $\"f%28x%29\"$ across all of its domain is $\"%22%5B+0+%2C%22\"$ $\"%22%2B%22\"$ $\"infinity\"$ $\"%22+%29%22\"$ . \n" ); document.write( "

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