I wouldn't give up.
Honestly, I never took pre-calculus in high school. I took up to Algebra II and then tested into Calculus when I took my first placement exam at my community college. I took pre-calculus anyway and had to put in a herculean effort to pass that class (my friend never studied and passed this class with flying colors, which made me a little jealous but you gotta focus on you and not what others have). I was also taking trigonometry and statistics at the time, which were a breeze. I wanted to do physics too!
I made it out with a B, but I put a decent amount of time into doing the homework (the homework was only worth 1% of our grade, the rest was based on tests).
I can't speak from experience from failing anything, but I recommend diving in and totally dedicating yourself to what your goals are. You may think you do that but you really have to look at your life and decide for certain what you want out of it. If there's a will, there's a way. I switched my major to math after that semester and I've been accepted to two universities as a math major and a third as a physics major. I remember sitting in my calc I class all excited and someone asked me how much math I needed and I said I had to go all the way. I did just that last semester and am glad to have made it. I don't know if you've ever heard of Richard Feynman, but I usually look to him for inspiration sometimes when physics gets a little boring or motivation comes by in small quantities. He's a very famous, non-stereotypical physicist.
Since you really need to take it again to avoid that F crushing your GPA, I recommend taking it one more time and really giving it your all, even if your "all" only gets you a C. I must've put in 5-6 hours a day because my teacher went nuts with her tests and I had to know everything inside and out when I was in pre-calculus. It was a good learning experience and it provided a good foundation for coming into calculus. Good algebra skills are essential to being good at higher math classes and physics.
Take advantage of everything. Go to office hours. Form study groups. Work ALL problems (do not use the solutions manual as a crutch, only a reference). < - Very important. Check out more textbooks and work those problems. Look for problems to do online. The only thing that is stopping you really is dedication.
I knew this kid during a summer about two years ago. I was ambitious and wanted to take Calc II as a summer class (8 weeks). Well, the kid (I shouldn't say kid) was a computer programmer and needed this for his major. He had taken it 5 times! When all was said and done, I saw him in the fall and I spoke with him about how he did. He passed with a C, but that is truly dedication because it really was the only thing stopping him from getting his degree. It was an insane amount of work to do and I basically lived and breathed integrals for that entire summer. I've had trouble getting my priorities straight lately since this is my last semester here, but you gotta know you can always turn things around, no matter how bleak! Hard work can accomplish a lot. I don't just mean doing your homework on time but believing in yourself and being extremely motivated to get what you want. I wouldn't give up. Absolutely know you did your best with no regrets when you leave a test. No "I could've studied harder".
I'm sure you can do it! I truly believe that high school doesn't represent much because I barely passed my math classes in high school and hated math back then. Look at me now, I'm majoring in it.