document.write( "Question 696709: find range ?
\n" ); document.write( " $\"sqrt%28-x%5E2%2B4x-3%29%2Bsqrt%28sin%28pi%2F2%2Asin%28pi%2F2%29%28x-1%29%29%29%29%29\"$ \n" ); document.write( "

Algebra.Com's Answer #429258 by mouk(232) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let
\n" ); document.write( " $\"+y+=+sqrt%28-x%5E2%2B4x-3%29+%2B+sqrt%28sin%28pi%2F2%2Asin%28pi%2F2%29%28x-1%29+%29%29+\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Then factorising the first term and tidying up the second gives:
\n" ); document.write( " $\"+y+=+sqrt%28%28-x%2B1%29%28x-3%29%29+%2B+sqrt%28sin%28pi%2F2%2A%28x-1%29+%29%29+\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "As we are assuming a real-valued function then both terms under the square root signs must be simultaneously positive (or zero).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Firstly consider $\"+%28-x%2B1%29%28x-3%29+%3E=+0+\"$
\n" ); document.write( " $\"+graph%28+200%2C+200%2C+-1%2C+4%2C+-10%2C+10%2C+-x%5E2%2B4x-3+%29+\"$
\n" ); document.write( "The solution to this is: $\"+1+%3C=+x+%3C=3+\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Secondly, consider $\"+sin%28pi%2F2%2A%28x-1%29%29+%3E=+0\"$
\n" ); document.write( " $\"+graph%28+200%2C+200%2C+-2pi%2C+2pi%2C+-3%2C+3%2C+sin%28pi%2F2%2A%28x-1%29%29+%29+\"$
\n" ); document.write( " $\"+sin%28pi%2F2%2A%28x-1%29%29+%3E=+0+\"$
\n" ); document.write( " $\"+0%3C=+pi%2F2%2A%28x-1%29+%3C=+pi+\"$
\n" ); document.write( " $\"+0%3C=+x-1+%3C=+2+\"$
\n" ); document.write( " $\"+1%3C=+x+%3C=+3+\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So in both cases, the square root terms simultaneously exist when $\"+1%3C=+x+%3C=+3+\"$ , and this is the range. \n" ); document.write( "

\n" );