Op-amp transfer function help

[Zero the hero]

Hi everyone,
I need to control an effect parameter by using a voltage varying form 3.78V to 4.10V. Since that's a very small range, passive resistor networks and voltage dividers didn't give a sufficient precision, stability and level of freedom when choosing components.
So I went into op-amps: I chose to feed an op-amp with a proper transfer function, so that when giving a voltage between 1V and 3V the op-amp returns a voltage between 3.78V and 4.10V, as the effect parameter needs.
I used this basic sketch, taken from "op-amps for everyone" by Texas Instruments:

I calculated these values for the resistors:
R1 = 33k
R2 = 4k7
RF = 33k
RG = 4k7
using
Vref  := +4.5V
Vout := +3.87V at Vin = 1v
Vout := +4.10V at Vin = 3v
I breadboarded everything, but it doesn't work... well, it actually returns a voltage, but it's the upper rail (8.12V using a regulated supply of 9V) of the op-amp, a TL072. An appropriate resistor divider and pot has been used for feeding the op-amp with the proper input voltage.
I noticed that disconnecting the Vref resistor (i.e. feeding no Vref to the circuit) the circuit returns a voltage output which is very small (+0.2V) or very big (upper rail), as if the circuit enlarges a lot the 1V .. 3V input voltage swing.

Now the questions...
What transfer function should I use? And what op-amp circuit sketch?
Is it a matter of Vref?
What kind of software could I use for simulating such simple circuits?
Any help is very appreciated!!!

Alessandro.

[Sir H C]

The op-amp is running as a buffer in this case (actually I think a gain of 2, I have to do the math...) .  You have a divider on the positive input then you gain back up at the output. 

You are best off with a resistor divider to take a wide range input to the reduced output range.  You need to do this with very precise references.  If you don't have precise references, then your very small range is never going to be hit.  I would do the divider with resistors and then run the op-amp as a buffer, then you get good accuracy and still have the range you want.

[Zero the hero]

Thanks for the suggestions!
Today I solved this problem: the picture inverted the resistor names so, once you solve the equation, you MUST swap R1 with R2 and RF with RG...
That's a pity, I wasted a complete day of work because of this.

[RaceDriver205]

I wasted a complete day of work because of this

Well hey, this is electronics! 

[Paul Perry (Frostwave)]

That's very good advice from Sir H C about using a voltage reference of some kind. It is easy to think "oh, the rail is 9v, I'll divide that down & subtract..." forgetting that the '9' might well be 7.5v or less one day.
The last device I made that had a similar section in it, I had the nominally 9v battery feed a low dropout 5v regulator & then go to a 7662 doubler. For that particular application, the regulated 5V was steady enough, but for anythink requiring real precision, then of course a reference diode is the go.
In norder of accuracy:
battery, plugpack, regulator, zener (with constant load), reference diode.