document.write( "Question 400822: what is the domain of √ ̅ (x-3) divided by x^2-8x+12? \n" ); document.write( "

Algebra.Com's Answer #283732 by robertb(5830) $\"\"$ $\"About$

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If your rational function is $\"sqrt%28%28x-3%29%2F%28x%5E2-8x%2B12%29%29+=+sqrt%28%28x-3%29%2F%28%28x-2%29%28x-6%29%29%29\"$ , then the domain is obtained as follows:
\n" ); document.write( "From the expression $\"%28x-3%29%2F%28x%5E2-8x%2B12%29+=+%28x-3%29%2F%28%28x-2%29%28x-6%29%29\"$ , the critical numbers are 2, 3, and 6. These critical numbers partition the real number line into the intervals ( $\"-infinity\"$ , 2), (2, 3), (3, 6), (6, $\"infinity\"$ ).
\n" ); document.write( "At ( $\"-infinity\"$ , 2), $\"%28x-3%29%2F%28%28x-2%29%28x-6%29%29+%3C+0\"$ by using the test point x = 0.
\n" ); document.write( "At (2, 3), $\"%28x-3%29%2F%28%28x-2%29%28x-6%29%29+%3E+0\"$ by using the test point x = 2.5.
\n" ); document.write( "At (3, 6), $\"%28x-3%29%2F%28%28x-2%29%28x-6%29%29+%3C+0\"$ by using the test point x = 4.
\n" ); document.write( "At (6, $\"infinity\"$ ), $\"%28x-3%29%2F%28%28x-2%29%28x-6%29%29+%3E+0\"$ by using the test point x = 7.
\n" ); document.write( "The critical numbers 2 and 6 are not included in the domain, as these will make the denominator equal to 0. We want those intervals that will make $\"%28x-3%29%2F%28%28x-2%29%28x-6%29%29+%3E+0\"$ , because the whole expression is under the square root symbol. The critical number x = 3 is included in the domain.\r
\n" ); document.write( "\n" ); document.write( "Hence the domain of the rational function is (2,3]U(6, $\"infinity\"$ ). \n" ); document.write( "

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