document.write( "Question 115062: Find the domain of f, the roots of f(x)=0, and determine the signs of f where\r
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\n" ); document.write( "\n" ); document.write( "f(x)= x^2-1/sqrt x+1 + sqrt x+1 \n" ); document.write( "

Algebra.Com's Answer #83696 by solver91311(24713) $\"\"$ $\"About$

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$\"f%28x%29=%28x%5E2-1%29%2F%28sqrt%28x%2B1%29%2Bsqrt%28x%2B1%29%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "This function is undefined if $\"x%2B1%3C=0\"$ , or putting a positive spin on the statement, the function is defined if and only if $\"x%2B1%3E0\"$ => $\"x%3E-1\"$ . Therefore, the domain of the function is the interval (-1, $\"infinity\"$ ]\r
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\n" ); document.write( "\n" ); document.write( "The zeros of the function can be seen readily if you factor the numerator and then rationalize the denominator.\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=%28%28x%2B1%29%28x-1%29%29%2F2sqrt%28x%2B1%29\"$
\n" ); document.write( " $\"f%28x%29=%28%28x%2B1%29%28x-1%29sqrt%28x%2B1%29%29%2F2sqrt%28x%2B1%29sqrt%28x%2B1%29\"$
\n" ); document.write( " $\"f%28x%29=%28%28x%2B1%29%28x-1%29sqrt%28x%2B1%29%29%2F%282x%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=0\"$ iff $\"x-1=0\"$ or $\"x%2B1=0\"$ or $\"sqrt%28x%2B1%29=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x-1=0\"$ iff $\"x=1\"$ , so $\"f%281%29+=+0\"$ and 1 is a zero of the function.
\n" ); document.write( " $\"x%2B1=0\"$ iff $\"x=-1\"$ , but -1 is not in the domain of the function, therefore $\"f%28-1%29\"$ is undefined (notice the open ended domain interval) and -1 is not a zero of the function.
\n" ); document.write( " $\"sqrt%28x%2B1%29=0\"$ iff $\"x%2B1=0\"$ => $\"x=-1\"$ , and again, -1 is not in the domain.\r
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\n" ); document.write( "\n" ); document.write( "Therefore the only zero of the function is 1.\r
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\n" ); document.write( "\n" ); document.write( "As to the balance of your question, \"determine the signs of f where...\" Where what? Re-post with the complete question and someone will try to answer. \n" ); document.write( "

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