let f = ( x, x2/1+x2) : x belongs to R be a function .Determine domain and range of the function.

please fast

We have,

Domain of f : Clearly, f(x) is defined for all x ∈ R as x2 + 1 ≠ 0 for any x ∈ R.

So, Domain (f) = R

Range of f : Let f(x) = y. Then,

 

⇒ x2 = x2y + y

⇒ x2 - x2y = y

⇒ x2(1- y) = y

 

 

Clearly, x will take real values, if

 

⇒ 0 ≤ y < 1

⇒ y ∈ [0,1)

Hence, range (f) = [0,1)

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