Question

Find range of the real function


`f = \left \{ \bigg(x, \: \frac{ {x}^(2) }{ {x}^(2) + 1} \bigg) : x \: \in \: R \right \}`
from R into R​.

Answer 1

Explanation:

`\large\underline{\sf{Solution-}}`

Given function is

`\rm \longmapsto\:f = \bigg(x, \: \frac{ {x}^(2) }{ {x}^(2) + 1} \bigg) : x \: \in \: R`

To find the range of the function f, Let assume that

`\rm \longmapsto\:y = \frac{ {x}^(2) }{ {x}^(2) + 1}`

`\rm \longmapsto\: {yx}^(2) + y = {x}^(2)`

`\rm \longmapsto\: {yx}^(2) - {x}^(2) = - y`

`\rm \longmapsto\: - {x}^(2) (1 - y) = - y`

`\rm \longmapsto\: {x}^(2) (1 - y) = y`

`\rm \longmapsto\: {x}^(2) = (y)/(1 - y)`

`\rm\implies \:x = \sqrt{(y)/(1 - y) }`

For x to be defined,

`\rm \longmapsto\:y \geqslant 0 \: \: and \: \: 1 - y > 0`

`\rm \longmapsto\:y \geqslant 0 \: \: and \: \: - y > - 1`

`\rm \longmapsto\:y \geqslant 0 \: \: and \: \: y < 1`

`\bf\implies \:y \: \in \: [0, \: 1)`

Hence,

`\bf\implies \:Range \: of \: f \: \in \: [0, \: 1)`