let f = ( x, x2/1+x2) : x belongs to R be a function Determine domain and range of - Maths - Relations and Functions

Question

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

Ankush Jain · Accepted Answer

We have,

$f (x) = \frac{x^{2}}{1 + x^{2}}$

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,

$f (x) = \frac{x^{2}}{1 + x^{2}} = y$

⇒ $\frac{x^{2}}{1 + x^{2}} = y$

⇒ x² = x²y + y

⇒ x² - x²y = y

⇒ x²(1- y) = y

⇒ $x^{2} = \frac{y}{1 - y}$

⇒ $x = � \sqrt{\frac{y}{1 - y}}$

Clearly, x will take real values, if

$\frac{y}{1 - y} \geq 0$

⇒ $\frac{y - 0}{y - 1} \leq 0$

⇒ 0 ≤ y < 1

⇒ y ∈ [0,1)

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

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