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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.

 Feb 11, 2023
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Finding the function F(x) that satisfies the given functional equation can be done through successive substitution. Here's one approach:

First, substitute x = 3 into the equation to get

F(3) + F(0) = 3.

Since F(0) is not defined, we'll replace it with a constant, say a. So, we have

F(3) + a = 3.

Next, substitute x = 2 into the equation to get

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

Substituting the value of F(3) from step 1 into this equation gives us

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

Now, substitute x = 1/2 into the equation to get

F(1/2) + F((2 * (1/2) - 3) / (1/2)) = 1/2.

Substituting the value of F(2) from step 2 into this equation gives us

F(1/2) + a = 1/2.

Finally, substituting the values of F(3) and F(1/2) from steps 1 and 3 into the original equation gives us

F(x) + a = x,

for all x ≠ 1 and x ≠ 2.

So, F(x) = x - a.

Since we have two equations in two unknowns (a and F(x)), we can solve for a. From step 1, a = 3 - F(3). And from step 3, a = 1/2 - F(1/2). Equating these two values of a gives us

3 - F(3) = 1/2 - F(1/2).

Substituting the expression for F(x) from step 4 into this equation gives us

3 - (3 - a) = 1/2 - (1/2 - a),

which simplifies to

a = 2.

So, the function F(x) that satisfies the given functional equation is

F(x) = x - 2.

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

 Feb 11, 2023

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