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Question

Determine whether function; f(x)=(1)[x] is even, odd or neither of the two (where [] denotes the greatest integer function).

A
f is an even function when xZ, f is an odd function when xZ
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B
f is an even function when xZ , f is an odd function when xZ
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C
f is an even function for all integers
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D
Neither odd nor even
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Solution

The correct options are
A f is an even function when xZ, f is an odd function when xZ
C f is an even function for all integers
D Neither odd nor even
Let f:RR be a function defined as f(x)=(1)[x].
When x is an integer [x]=x, hence f(x)=(1)[x]=(1)x, which is an even function.
If x is a positive real number which is not an integer, then [x] is the integer part of the given number, say a.
Therefore f(x)=±1 based on a is even or odd.
If x is a negative real number which is not an integer, then [x]=a1, where a is the integer part of the given number.
Therefore f(x)=±1 based on a is odd or even.
Hence f(x)=f(x) when xZ

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