Which of the following options is-are true for the Greatest integer function -x- x lt-x-1x-1 lt-x- x-x-m-x-m- if m- nbsp-x-m-x-m- if m-

The correct options are A [x]≤ x lt;[x]+1 B x 1 lt;[x]≤ x D [x+m]=[x]+m, if m∈ℤWe know that, nbsp; The real function f:ℝ→ℝ defined by f(x)=[x] takes value ...

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Standard XII

Mathematics

Greatest Integer Function

Which of the ...

Question

Which of the following options is/are true for the Greatest integer function [.]?

[x] \leq x < [x] + 1

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x - 1 < [x] \leq x

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[x + m] = x + [m],

m \in Z

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[x + m] = [x] + m,

m \in Z

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Solution

The correct options are
A $[x] \leq x < [x] + 1$
B $x - 1 < [x] \leq x$
D $[x + m] = [x] + m,$ if $m \in Z$
We know that,
The real function $f : R \to R$ defined by $f (x) = [x]$ takes value less than or equal to $x$

So,
$[x] = x$ if $x$ is an integer
$\Rightarrow [x] \leq x < [x] + 1$
$\Rightarrow x - 1 < [x] \leq x$
$\Rightarrow [x + m] = [x] + m,$ if $m \in Z$

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Similar questions

The relation $x^{m} \div x^{n} = x^{m - n}$ is true, if $x > 0$ and $m \leq n$ .

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Then

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Greatest Integer Function

Standard XII Mathematics

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Which of the following options is/are true for the Greatest integer function [.]?

The correct options are A [x]≤x<[x]+1 B x−1<[x]≤x D [x+m]=[x]+m, if m∈ZWe know that, The real function f:R→R defined by f(x)=[x] takes value less than or equal to x So, [x]=x if x is an integer ⇒[x]≤x<[x]+1 ⇒x−1<[x]≤x ⇒[x+m]=[x]+m, if m∈Z