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# DUDE

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Mar 5, 2023

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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.

Mar 5, 2023