Question

Consider f : R + → [4, ∞ ) given by f ( x ) = x 2 + 4. Show that f is invertible with the inverse f −1 of given f by , where R + is the set of all non-negative real numbers.

Answer 1

The function provided is f( x )= x 2 +4 for the range f: R + →[4,∞) .

For the function to be invertible, the function should be one-one and onto.

The equation to check whether the provided function is one-one is,

f( x )=f( y ) x 2 +4= y 2 +4 x 2 = y 2 x=y [x≠−y as the domain is R + ]

Therefore, every element in the domain has a unique image, and the function is a one-one function.

Now to check whether the function is onto:

Consider a function y∈[4,∞) , such that,

y= x 2 +4 x 2 =y−4 x= y−4 ≥0 g( y )= y−4

So, for any y∈R there exists x= y−4 ∈R in such a way that,

f( x )=f( y−4 ) = ( y−4 ) 2 +4 =y−4+4 =y

As the particular value of the function exist in the range provided, so the function is onto.

Thus, the f function is one-one and onto, so f −1 exists.

( gof )( x )=g( f( x ) ) =g( x 2 +4 ) = ( x 2 +4 )−4 =x

( fog )( y )=f( g( y ) ) =f( y−4 ) = ( y−4 ) 2 +4 =y .

Therefore,

gof= I R fog= I R fog=gof

So the function f is invertible.

y= x 2 +4 x= y−4 g( y )= y−4 f −1 ( y )= y−4

Thus, the inverse of the function is f −1 ( y )= y−4 .