# CAT Practice : Functions

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The greatest integer function is one of the most interesting around. Input can be anything, output is discrete - this function is a logician's delight. You can create gazillion questions; none of which can be solved by plugging in formulae.

## Greatest Integer Function

Q.5: [x] = greatest integer less than or equal to x. If x lies between 3 and 5,5 inclusive, what is the probability that [x2] = [x]2?
1. Roughly 0.64
2. Roughly 0.5
3. Roughly 0.14
4. Roughly 0.36

Choice C. Roughly 0.14

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## Detailed Solution

Let us take a few examples.
[32] = [3]2
[3.52] = 12 [3.5]2 = 9
[42] = 16 [4]2 = 16
For x ∈ (3, 5). [x]2 can only take value 9, 16 and 25.

Let us see when [x2] will be 9, 16 or 25.
If [x2] = 9,
x2 ∈ [9, 10)
=> x ∈ [3, $\sqrt {{\rm{10}}}$)

[x2] = 16
x2 ∈ [16, 17)
=> x ∈ [4, $\sqrt {{\rm{17}}}$)

In the given range [x2] = 25 only when x = 5
So [x2] = [x]2 when x ∈ [3, $\sqrt {{\rm{10}}}$] or [4, $\sqrt {{\rm{10}}}$) or 5.
Probability = ${{\sqrt {10} - 3 + \sqrt {{\rm{17}}} - 4} \over 2}$
${{3.16 - 3 + 4.12 - 4} \over 2} = {{0.28} \over 2}$ = 0.14

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Functions.