CARJ FLT CREW LOSES SITUATIONAL AWARENESS ON DEP FROM PSP.

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WE WERE CLRED FOR THE PALM SPRINGS 4 DEP. AFTER TKOF ON RWY 31L; WE COMPLIED BY HDG 309 DEGS. SHORTLY AFTER TKOF; WE WERE GIVEN DIRECT TNP. I WAS THE PNF AND SO I INPUTTED IT IN THE FMS AND AFTER VERIFICATION FROM THE PF; I EXECUTED IT. THE CLRNC FROM ATC WAS A R TURN TO TNP. AFTER VERIFYING IT; WE REALIZED THAT WOULD BE A 270 DEG TURN. I ASKED THE CTLR TO CONFIRM A 'R TURN TO TNP.' HE REITERATED IT. BECAUSE WE HAD HAD PROBS WITH THE FMS ON OUR INBOUND FLT; I SWITCHED TO 'GREEN NEEDLES' WHICH IS MANUAL NAV AND DIALED IN TNP. THERE WAS NO RECEPTION (WE WERE TOO CLOSE I THINK). MEANWHILE; I WAS RUNNING THE AFTER TKOF CHKLIST. WE NOTICED A MESSAGE ON THE FMS STATING GPS DR. ATC INQUIRED ABOUT OUR DIRECTION OF FLT. I RESPONDED WITH OUR HDG AND ASKED FOR A HDG TO TNP. HE GAVE ABOUT A 060 DEG HDG. I BELIEVE WE WERE HDG ABOUT 130 DEGS. I THEN EXPLAINED THE PROBS WE HAD BEEN HAVING WITH OUR FMS AND TOLD HIM TO RE-FILE US AS A NON-GPS ACFT (ONLY VOR/DME/TRANS). WE RETURNED TO SLC WITH NO FURTHER EVENTS. THE MECHS SWITCHED THE ENTIRE FMS COMPUTER. CONTRIBUTING FACTORS WERE: NEW ARPT FOR BOTH PLTS; UNFAMILIARITY WITH FMS ERROR CODES (GPS DR; DR EXCEEDS 5 MINS); BUSY TIME OF FLT (TKOF; CLB) UNFAMILIARITY OF AREA (WE COULD HAVE HAD BETTER SITUATIONAL AWARENESS); FAULTY EQUIP (MAINT SHOULD HAVE DEFERRED THE FMS INSTEAD OF RESETTING IN PALM SPRINGS); AND ATC COULD HAVE GIVEN US A HDG TO TNP (IE; HDG 060 DEGS; WHEN ABLE DIRECT TNP). WE WOULD HAVE KNOWN IMMEDIATELY. FMS'S ARE SO RELIABLE WE ARE NOT ACCUSTOMED TO OPERATING WITHOUT IT. WE ARE ALSO NOT USED TO IT SHOWING ERRONEOUS INFO. TNP WAS 60 OR SO MI NE OF WHERE OUR FMS THOUGHT IT WAS. APPARENTLY THIS IS KNOWN AS MAP SHIFT. FORTUNATELY; IT WAS VFR AND WE COULD MAINTAIN TERRAIN CLRNC AT ALL TIMES. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATED THAT THE 130 DEG HDG AT THE TIME ATC QUERIED THEM WAS THE RESULT OF HAVING COMPLETED 180 DEGS OF THE 270 DEG R TURN THE FMS WAS INDICATING. WHEN ADVISED OF THE ACTUAL RELATIVE POS OF TNP; THE FLT CREW DETERMINED THEY COULD NOT TRUST THE FMS PRESENTATION FOR REASONS THE WERE UNABLE TO DETERMINE. AS A RESULT; THEY TRANSITIONED TO RAW DATA VOR NAV AND WROTE UP THE FMS AS FAILED. THE RPTR HAD NOT PROGRAMMED THE FMS NOR HAD SHE REVIEWED THE OTHER PLT'S PROGRAMMING OF IT. SHE ALSO DOES NOT REMEMBER ACTUALLY SEEING A RWY SYMBOL ON THE NAV DISPLAY ON TKOF. AS A RESULT RPTR COULD NOT DISPUTE THE POSSIBILITY OF A FAULTY INITIALIZATION OF THE ACFT POS. RPTR FELT IT WAS AT LEAST POSSIBLE THAT AN INCORRECT 'PRESENT POS' HAD BEEN PROGRAMMED INDICATING THE ACFT WAS N OF ITS ACTUAL POS THUS RESULTING IN THE INACCURATE DEPICTION OF TNP.