Q. 134.9( 7 Votes )

Consider f: R<

Answer :

f: R+ [4, ] given by f(x) = x2 + 4


One-one:


Let f(x) = f(y)


x2 + 4 = y2 + 4


x2 = y2


So, x = y [as x = y R]


Therefore, f is a one-one function.


Onto:


For y [4, ∞), let y = x2 + 4


x2 = y – 4 0 [as y 4]



Therefore, for any y R, there exists such that



So, f is onto.


Thus, f is one-one and onto and therefore, f-1 exists.


Let us define g: [4, ∞) R+ by,



Now,




So, gof = fog = IR+


Hence, f is invertible and the inverse of f is given by



Hence, proved.


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