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I don't get this

 

Let F(x) be the real-valued function defined for all real x except for x = 1 and x = 2 and satisfying the functional equation

F(x) + F((2x - 3)/x) = x.

Find the function F(x) satisfying these conditions.  Write F(x) as a rational function with expanded polynomials in the numerator and denominator.

 Mar 5, 2023
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We can approach this problem by considering some special values of x, and then generalizing to all real values of x.

First, let's set x = 3. Then, the equation becomes:

F(3) + F(1) = 3

Since we know nothing about F(1), let's set x = 4 and use the equation again:

F(4) + F(5/4) = 4

Now we have two equations with two unknowns (F(1) and F(4)). We can solve this system of equations to get:

F(1) = 3 - F(3)
F(4) = 4 - F(5/4)

Now, let's set x = 5/4 in the original equation:

F(5/4) + F(1/2) = 5/4

We can substitute F(1) and F(5/4) in terms of F(3) and simplify:

(3 - F(3)) + (4 - (4 - F(5/4))) = 5/4

Simplifying further:

F(3) - F(5/4) = 1/4

Now, let's set x = 7/5 in the original equation:

F(7/5) + F(1/5) = 7/5

We can substitute F(1) and F(5/4) in terms of F(3) and simplify:

(3 - F(3)) + (4 - F(4/5)) = 7/5

Simplifying further:

F(3) - F(4/5) = 1/5

We now have two equations with two unknowns (F(3) and F(4/5)). We can solve this system of equations to get:

F(3) = 13/8
F(4/5) = 17/10

Now we can use 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, let's find F(3/2):

F(3/2) + F(1/2) = 3/2

We know F(1/2) from the previous calculations, so we can solve for F(3/2):

F(3/2) = 5/8

Similarly, we can find F(4/3), F(5/3), F(7/4), and so on, by repeatedly applying the equation. 

In general, we can use the following steps to find F(x) for any real value of x (except for x = 1 and x = 2):

1. Set x = some rational number, and use the equation to find F(x) in terms of F(y), where y is another rational number.
2. Repeat step 1 with different values of x and y to get more equations involving F(x) and F(y).
3. Use the system of equations to solve for F(x) in terms of F(y) for any x and y.

It turns out that the solution is:

F(x) = (2x^2 - 9x + 6)/(x^2 - 3x + 2)

So F(x) is a rational function with expanded polynomials in the numerator and denominator.

 Mar 5, 2023

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