Question

Y=1/x²+2. Find the domain and range but plz write down every step and reason.

Answer 1

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

$\sf \purple{domain}$

$\sf since \: there \: are \: no \: restrictions \: for \: funtion \: to \: \\ \sf \: exists.Domain ∈ \blue\R$

$\sf \purple{range}$

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

$\implies {x}^{2} + 2= \frac{1}{ y }$

$\implies {x}^{2} = \frac{1}{ y } - 2$

$\implies x = \sqrt{ \frac{1}{ y } - 2}$

$\implies \frac{1}{y} - 2 > 0$

$\implies \frac{1}{y} > 2$

$\implies y < \frac{1}{2}$

$\sf and \: y ≠0$

$\therefore y∈( - ∞,0)∪(0, \frac{1}{2} )$

$\orange{ \therefore \sf range \: ∈( - ∞,0)∪(0, \frac{1}{2} )}$

HOPE IT HELPS!!