If f(x)=2(x+1) (x+1) + 3 for x greater than equal to k has inverse , what is minimum value of k?

Question

aavas

Aavas

 

Aavas

Aavas

Aavas

Answer 1

To find the inverse let y=f(x)=(x+1)^2+3. Therefore (x+1)^2=y-3 and x+1=√(y-3). Let g(x)=√(x-3)-1 then g(x) is the inverse of f(x). The inverse can only exist if x≥3. f(g(x))=x by definition of inverse so f(g(x))≥k.

g(x) is minimum when x=3 and g(3)=-1 and f(-1)=3 so for g(x) to exist, x≥-1; therefore k=-1.