Hi,
For the NHE E1 https://imgur.com/a/QRDC8
How do we also get c>2 as a solution too?
A way to look at this question, though not intuitive in the slightest may be to consider the graph
Here we have the solutions to our inverse function, which you have clearly understood.
Looking at the graphs we know that c moves g(x) left and right. We know that moving it left will generate solutions if we start at c=0. We also know that the largest (or smallest, depending on the case) value of c in which there will be two solutions is the c value when
Solving for this we get
and
Thus we have
and
You must also consider that there is most certainly more than one value in which the inverses only intercept once, and this should be a trigger to consider some creative options.
EDIT:
Upon further inspection, I think I may be able to offer a more intuitive solutions:
Consider the function
It has a stationary point at
As our square root function is positive we will take the positive band of this quadratic. Thus our domain for our inverse will be
Rearranging the function g we can obtain the following
From this we can state that
From here, there will be one solution when these two x values are the same or when they intersect in the allowable domain. That is
And
Solving for this we get
and