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Maths-

General

Easy

Question

If $f colon R not stretchy rightwards arrow R text is defined by end text f left parenthesis x right parenthesis equals x minus left square bracket x right square bracket minus 1 half text fo end text x$ where $left square bracket x right square bracket$ is the greatest integer no exeeding $x comma text then end text open curly brackets x element of R colon f left parenthesis x right parenthesis equals 1 half close curly brackets equals$

$Z$ , the set of all integers
$N$ , the set of all natural numbers
$\emptyset$ , the empty set
$R$

hint Hint:

To find x when f(x)= $1 half$ for the given function, first we know that $x equals open square brackets x close square brackets plus open curly brackets x close curly brackets rightwards double arrow open curly brackets x close curly brackets equals x minus open square brackets x close square brackets$ , we can write f(x)= $open curly brackets x close curly brackets minus 1 half$ , now will find the condition when f(x) will be $1 half$ .

The correct answer is: $straight empty set$ , the empty set

Given, f(x)= $x minus open square brackets x close square brackets minus 1 half$
we know that, x= $open square brackets x close square brackets plus open curly brackets x close curly brackets$
$x minus open square brackets x close square brackets equals open curly brackets x close curly brackets$
Now, f(x) = $open curly brackets x close curly brackets minus 1 half$
For, $f left parenthesis x right parenthesis equals 1 half$ $open curly brackets x close curly brackets minus 1 half equals fraction numerator 1 over denominator 2 space end fraction open curly brackets x close curly brackets equals 1 space comma space w h i c h space i s space n o t space p o s s i b l e space a s space open curly brackets x close curly brackets space i s space a space f r a c t i o n a l space p a r t space f u n c t i o n comma space 0 less or equal than open curly brackets x close curly brackets less than 1 S o comma f o r space f left parenthesis x right parenthesis equals 1 half comma space space x equals empty set$

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