+195 Sure! Here are the steps to find the function F(x) in detail:
1. Set x = 3 in the equation F(x) + F((2x - 3)/x) = x. We get:
F(3) + F(1) = 3
2. Set x = 4 in the same equation. We get:
F(4) + F(5/4) = 4
3. Use the equation F(3) + F(1) = 3 to solve for F(1) in terms of F(3):
F(1) = 3 - F(3)
4. Use the equation F(4) + F(5/4) = 4 to solve for F(4) in terms of F(5/4):
F(4) = 4 - F(5/4)
5. Substitute F(1) and F(5/4) in terms of F(3) into the equation F(5/4) + F(1/2) = 5/4. We get:
(3 - F(3)) + (4 - (4 - F(5/4))) = 5/4
Simplifying:
F(3) - F(5/4) = 1/4
6. Substitute F(1) and F(5/4) in terms of F(3) into the equation F(7/5) + F(1/5) = 7/5. We get:
(3 - F(3)) + (4 - F(4/5)) = 7/5
Simplifying:
F(3) - F(4/5) = 1/5
7. Solve the system of equations F(3) - F(5/4) = 1/4 and F(3) - F(4/5) = 1/5 for F(3) and F(4/5). We get:
F(3) = 13/8
F(4/5) = 17/10
8. Substitute F(4/5) and F(3) into the equation F(3/2) + F(1/2) = 3/2. We get:
F(3/2) = 5/8
9. Generalize the pattern by using the equation F(x) + F((2x - 3)/x) = x to find F(x) for any real value of x (except for x = 1 and x = 2). For example, to find F(4/3), we have:
F(4/3) + F(1/3) = 4/3
Substitute F(3) and F(1/3) into the equation:
F(3) + F(5/3) = 4/3
Solve for F(5/3):
F(5/3) = -1/24
10. Use the same method to find F(x) for any real value of x (except for x = 1 and x = 2).
11. The solution is:
F(x) = (2x^2 - 9x + 6)/(x^2 - 3x + 2)
Therefore, F(x) is a rational function with expanded polynomials in the numerator and denominator.
