Question: How to make this plot work?

I have an equation which has no explicit solutions, but I would like to plot the solutions. I have construction a function which, for every argument, produces a numerical answer to the equation, but this function can't be plotted. Below is a simplified (equation has trivial explicit solution) function which shows the difficulty.

> Fe := p->fsolve(eval(1-x+p = x), x)
p -> fsolve(eval(1 - x + p = x), x)

Here we exercise the function to produce floating results.
> Fe(0) = 0.5000000000
> Fe(.5) = 0.7500000000
> Fe(1) = 1.

Now try to plot the function.
> plot(Fe(a), a = 0 .. 1)
Error, (in fsolve) a is in the equation, and is not solved for

Is there any way to plot the function as defined above? Or, is there any way to modify the function to make it plotable?

Thanks,

Dave