document.write( "Question 1196650: Determine whether function;f(x) =(-1)^[x] is even,odd or neither of two (where[]denotes the greatest integer function). \n" ); document.write( "

Algebra.Com's Answer #829570 by MathLover1(20849) $\"\"$ $\"About$

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$\"f%28x%29+=%28-1%29\"$ ^[x] is even, odd or neither \r
\n" ); document.write( "\n" ); document.write( "Let $\"f\"$ : $\"R\"$ → $\"R\"$ be a function defined as $\"f%28x%29=%28-1%29\"$ ^[x]
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\n" ); document.write( "When $\"x+\"$ is an integer [x]= $\"x\"$ , hence $\"f%28x%29=%28-1%29\"$ ^[x] = $\"%28-1%29%5Ex\"$ , which is an even function.\r
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\n" ); document.write( "\n" ); document.write( "If $\"x\"$ is a positive real number which is not an integer, then [x] is the integer part of the given number, say $\"a\"$ .\r
\n" ); document.write( "\n" ); document.write( "Therefore $\"f%28x%29\"$ =± $\"1\"$ based on $\"a\"$ is even or odd.\r
\n" ); document.write( "\n" ); document.write( "If $\"x+\"$ is a negative real number which is not an integer, then [x]= $\"a%5E-1\"$ , where $\"a\"$ is the integer part of the given number.\r
\n" ); document.write( "\n" ); document.write( "Therefore $\"f%28x%29\"$ =± $\"1\"$ based on $\"a\"$ is $\"odd\"$ or $\"even\"$ .\r
\n" ); document.write( "\n" ); document.write( "Hence $\"f%28x%29=-f%28-x%29\"$ when $\"x\"$ not ∈ $\"Z+\"$
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\n" ); document.write( "\n" ); document.write( "so, correct answers are:
\n" ); document.write( " $\"f%28x%29\"$ is an $\"even\"$ function when $\"+x\"$ ∈ $\"+Z\"$
\n" ); document.write( " $\"f%28x%29\"$ is an $\"odd+\"$ function when $\"x\"$ is $\"not\"$ ∈ $\"Z\"$
\n" ); document.write( " $\"f%28x%29\"$ is $\"neither+\"$ even nor odd\r
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