Question

If g:(2,0) = (2,00) is defined by g(x) Show that the function g is one-to- X-2 one and onto.

Answer 1

Given 9: (2,00) → (2.0), defined by n+1 n-2 Now g (m)s (n-2)+2+1 (1-2) n-2 + 3 0-2

Now to prove one-one'. Lete take non E (210) with Tu t'u Now z tu & zou Since 1 & 2 & 12 ㅋ e-lu 12-2 then It 3 E # t + 7,-2 2-2

That is $g(x_{1}) \neq g(x_{2}) .$ Therefore g is injective.

Now To prove onto'. is onto. Suppore then for for any E (2,00), there existe ne (20) such that y=80) ntl y = 1+ 3 n-2 n-2 3 Y-1 n-2 n-2 3 1-23 y-

M-2 = _3 S-1 M Elw +2.

Since $y \in (2, \infty),$

$y>2 \Rightarrow y-1> 1.$

le 170. Now I 0 3 70 91 thg zt ㅋ 2 Now. since 1 then JI+0. 1e Lo then 이공 뙤 +2Loo ㅋ 40

thus 24 n 1 so le 1 6 (29)

Thus our assumption is right, that is, for any y in the given range, there exists x in tge the domain such that

g(x)=y.

That is g is onto.