Question

Find the domain and range of the function f(x)= 1/sqrt(x-[x])

Answer 1

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Answer 2

it is given that function, f(x) = $\frac{1}{\sqrt{x-[x]}}$ where [.] is greatest integer.

we have to find the domain and range of the given function.

for square root to be defined, x - [x] ≥ 0

but x - [x] ≠ 0

so, x - [x] > 0

⇒x > [x]

this is true for all real value of x except x = I

so, x ∈ R - {I} hence domain ∈ R - {I}

now f(x) = 1/√{x - [x]}

we know, {x} = x -[x] where {.} is fractional part of x.

so, f(x) = 1/√{x}

we know, 0 ≤ {x} < 1

⇒0 ≤ √{x} < 1

⇒1 < 1/√{x} < ∞

hence range of the function, y ∈ (1, ∞)