Question 1028867
Look first for the inverse relation to f.


{{{(R(x)-4)^2+1=x}}} for the inverse function of f.
{{{(R(x)-4)^2=x-1}}}
{{{R(x)-4=0+- sqrt(x-1)}}}
{{{R(x)=4+- sqrt(x-1)}}}


The domain of R(x) is for TWO different functions.  R itself is a relation and not a function.  This is because the "plus or minus" part of the expression.  The domain for either branch of R  is  {{{x>=4}}}.


Look again at function f(x).  Domain is ALL REAL NUMBERS.  What about the range of f(x)?   f(x) is a parabola with a vertex minimum value at  (4,1).  This means that the RANGE for f(x)  is  {{{x>=4}}}, or as a description, from 4 inclusive toward infinity.  


Going from a function  (f(x)) to its inverse, the domain and range switch.  The range {{{x>=4}}}  for f(x)  is the DOMAIN for either branch of R(x).




Function f(x)
{{{graph(300,300,-3,12,-3,12,(x-4)^2+1)}}}


Upper branch of R(x)
{{{graph(300,300,-3,12,-7,8,-4+sqrt(x-1))}}}


Lower branch of R(x)
{{{graph(300,300,-3,12,-7,8,-4-sqrt(x-1))}}}