Question

Consider f : R → R given by f ( x ) = 4 x + 3. Show that f is invertible. Find the inverse of f .

Answer 1

The function provided is f( x )=4x+3 for the range f:R→R .

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

f( x )=f( y ) 4x+3=4y+3 4x=4y x=y

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

Consider a function y∈R , such that,

y=4x+3 x=( y−3 4 ) g( y )= y−3 4

So, for any y∈R there exists x= y−3 4 ∈R in such a way that,

f( x )=f( y−3 4 ) =4( y−3 4 )+3 =y

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

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

( gof )( x )=g( f( x ) ) =g( 4x+3 ) = ( 4x+3 )−3 4 =x

( fog )( y )=f( g( y ) ) =f( y−3 4 ) =4( y−3 4 )+3 =y .

Therefore,

gof= I R fog= I R fog=gof

So the function f is invertible.

y=4x+3 x=( y−3 4 ) g( y )= y−3 4 f −1 ( y )= y−3 4

Thus, the inverse of the function is f −1 ( y )= y−3 4 .