+4 Let be the real-valued function defined for all real except for and and satisfying the functional equation
Find the function \(F(x) + F \left( \frac{2x - 3}{x - 1} \right) = x\) satisfying these conditions. Write as a rational function with expanded polynomials in the numerator and denominator.
I saw a question with a similar structure https://web2.0calc.com/questions/help-for-functional-equation but for this question the computations get very large.
Thank you for your help!
+1768 This functional equation can be solved to find f(x) as a rational function. Here's how:
Isolating f(x): We want to isolate f(x) on one side of the equation. Subtract f((2x - 3)/(x - 1)) from both sides:
f(x) = x - f((2x - 3)/(x - 1))
Finding a Common Denominator: The next step is to find a common denominator for the right side of the equation. Since the function is undefined at x = 1 and x = 2, the common denominator will involve these terms (x - 1) and (x - 2).
We can rewrite f((2x - 3)/(x - 1)) as:
f((2x - 3)/(x - 1)) * (x - 2) / (x - 2) = f((2x - 3)/(x - 1)) * (x - 2) / (x - 1)(x - 2)
This introduces the term (x - 2) in the denominator to match the existing term (x - 1).
Substituting and Combining Like Terms:
Now, substitute the entire right side of the equation back into the expression for f(x):
f(x) = x - [ f((2x - 3)/(x - 1)) * (x - 2) / (x - 1)(x - 2) ]
Expand the product in the denominator:
f(x) = x - [ f((2x - 3)/(x - 1)) * (x - 2) ] / (x^2 - 3x + 2)
Recognizing a Pattern:
Notice that the term f((2x - 3)/(x - 1)) appears in the equation. We can substitute the entire right side of the original equation for this term:
f(x) = x - { [ x - f((2x - 3)/(x - 1)) ] * (x - 2) } / (x^2 - 3x + 2)
This creates a recursive relationship involving f(x). However, it allows us to eliminate f((2x - 3)/(x - 1)) from the equation entirely.
Simplifying and Solving:
Expand the product in the brackets and combine like terms:
f(x) = x - (x^2 - 5x + 2) / (x^2 - 3x + 2)
f(x) = (x^2 - 3x + 2 - x^2 + 5x - 2) / (x^2 - 3x + 2)
f(x) = 2x / (x^2 - 3x + 2)
Therefore, the function f(x) satisfying the given conditions is:
f(x) = 2x / (x^2 - 3x + 2)
