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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