Question

What will be the inverse of f(x)=(x+1)²-1 on R->R?
I think that the function is not bijective as it is not one one and onto.
Can you please help me with this?
Thanks.

Answer 1

Domain of f(x) : (-inf, inf)
Range of f(x) : [-1, inf )

As you said, This is not bijective as for every element y (except -1) in the range, There are two elements (x) in domain that maps to y.

So unless , We restrict the domain, there can't be any inverse.

So Let us try restricting.

y = (x+1)^2 -1

x= sqrt(y+1) - 1. We only take positive value of sqrt(y+1).

We know , y>=-1

So (y+1) >=0

So, x =sqrt(y+1)-1 >= -1

Hence x belongs to [-1, inf).

So , If we redefine the domain to be [ -1, inf) and range is also [-1, inf) , The function has inverse and is given by sqrt(x+1) - 1