The number of integers belonging to the interval -3-30- but not belonging to the range of the function fx-x x -x-x-x- R- iswhere - - - and - denote the greatest integer function and fractional part respectively-

Consider only integers in this case #160; f(x)=x{x } x[ x]=x(x [x]) x[ x]=x^2 x([x]+[ x])=x^2 x∈ [ 3,30] f(x)∈ [0,900] The integers #160; belonging ...

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Standard XII

Mathematics

Definition of Function

The number of...

Question

The number of integers belonging to the interval $[- 3, 30]$ but not belonging to the range of the function
$f (x) = x {x} - x [- x], x \in R$ , is
(where $[.]$ and ${.}$ denote the greatest integer function and fractional part respectively.

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Solution

Consider only integers in this case
$f (x) = x {x} - x [- x] = x (x - [x]) - x [- x] = x^{2} - x ([x] + [- x]) = x^{2}$
$x \in [- 3, 30] ⟹ f (x) \in [0, 900]$
The integers belonging to the range are $0, 1, 4, 9, 16, 25$
Number of integers does not belonging to range are $34 - 6 = 28$

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The number of integers belonging to the interval [−3,30] but not belonging to the range of the functionf(x)=x{x}−x[−x],x∈R, is(where [.] and {.} denote the greatest integer function and fractional part respectively.

Consider only integers in this case f(x)=x{x}−x[−x]=x(x−[x])−x[−x]=x2−x([x]+[−x])=x2x∈[−3,30]⟹f(x)∈[0,900]The integers belonging to the range are 0,1,4,9,16,25Number of integers does not belonging to range are 34−6=28

The number of integers belonging to the interval $[- 3, 30]$ but not belonging to the range of the function
$f (x) = x {x} - x [- x], x \in R$ , is
(where $[.]$ and ${.}$ denote the greatest integer function and fractional part respectively.

Consider only integers in this case
$f (x) = x {x} - x [- x] = x (x - [x]) - x [- x] = x^{2} - x ([x] + [- x]) = x^{2}$
$x \in [- 3, 30] ⟹ f (x) \in [0, 900]$
The integers belonging to the range are $0, 1, 4, 9, 16, 25$
Number of integers does not belonging to range are $34 - 6 = 28$