we have been given a quadratic function
and we need to restrict the domain such that it becomes a one to one function.
we know that vertex of this quadratic function occurs at (5,2).
further, we know that range of this function is
.
if we restrict the domain of this function to either
or
, it will become one to one function.
let us know find its inverse.
![y=(x-5)^{2}+2](/tex.php?f=y=(x-5)^{2}+2)
upon interchanging x and y, we get:
![x=(y-5)^{2}+2](/tex.php?f=x=(y-5)^{2}+2)
let us now solve this function for y.
![(y-5)^{2}=x-2\\ y-5=\pm \sqrt{x-2}\\ y=5\pm \sqrt{x-2}\\](/tex.php?f=(y-5)^{2}=x-2\\ y-5=\pm \sqrt{x-2}\\ y=5\pm \sqrt{x-2}\\)
hence, the inverse function would be
if we restrict the domain of original function to
and the inverse function would be
if we restrict the domain to
.