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  4. If f (x)=[x] and g(x)=x-[x], then which of the following is the zero function?

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Q. If $f \left(x\right)=\left[x\right]$ and $g\left(x\right)=x-\left[x\right]$, then which of the following is the zero function?

Relations and Functions

Solution:

$\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)=\left[x\right]+\left(x-\left[x\right]\right)=x$
$\left(fg\right)\left(x\right)=f\left(x\right). g\left(x\right)=\left[x\right]\left(x-\left[x\right]\right)$
$\left(f-g\right)\left(x\right)=f\left(x\right)-g \left(x\right)=\left[x\right]-\left(x-\left[x\right]\right)=2\left[x\right]-x$