SOLUTION: The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies xf(x) + f(1 - x)/x = x^3 + 3x^2 + 14x - 13 for all real x. Find f(x).

Algebra ->  Functions -> SOLUTION: The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies xf(x) + f(1 - x)/x = x^3 + 3x^2 + 14x - 13 for all real x. Find f(x).      Log On


   

Question 1209971: The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies
xf(x) + f(1 - x)/x = x^3 + 3x^2 + 14x - 13
for all real x. Find f(x).
Answer by CPhill(1959) About Me  (Show Source):
You can put this solution on YOUR website!
Absolutely, let's solve this functional equation.
**Understanding the Problem**
We're given a functional equation that relates f(x) and f(1 - x). Our goal is to find an explicit expression for f(x).
**Solving the Functional Equation**
1. **Original Equation:**
$$xf(x) + \frac{f(1 - x)}{x} = x^3 + 3x^2 + 14x - 13 \quad (*)$$
2. **Substitute (1 - x) for x:**
Replace every instance of 'x' with '(1 - x)' in the original equation:
$$(1 - x)f(1 - x) + \frac{f(1 - (1 - x))}{1 - x} = (1 - x)^3 + 3(1 - x)^2 + 14(1 - x) - 13$$
Simplify:
$$(1 - x)f(1 - x) + \frac{f(x)}{1 - x} = (1 - 3x + 3x^2 - x^3) + 3(1 - 2x + x^2) + 14 - 14x - 13$$
$$(1 - x)f(1 - x) + \frac{f(x)}{1 - x} = 1 - 3x + 3x^2 - x^3 + 3 - 6x + 3x^2 + 14 - 14x - 13$$
$$(1 - x)f(1 - x) + \frac{f(x)}{1 - x} = -x^3 + 6x^2 - 23x + 5 \quad (**)$$
3. **Multiply (*) by (1 - x):**
Multiply the original equation (*) by (1 - x):
$$x(1 - x)f(x) + \frac{(1 - x)f(1 - x)}{x} = (1 - x)(x^3 + 3x^2 + 14x - 13) \quad (***)$$
4. **Multiply (**) by x:**
Multiply the substituted equation (**) by x:
$$x(1 - x)f(1 - x) + f(x) = x(-x^3 + 6x^2 - 23x + 5) \quad (****)$$
5. **Solve for f(1-x) in (***):**
$$(1-x)f(1-x) = x(x^3 + 3x^2 + 14x - 13) - x^2(1-x)f(x)$$
Divide by (1-x):
$$f(1-x) = \frac{x(x^3 + 3x^2 + 14x - 13) - x^2(1-x)f(x)}{1-x}$$
6. **Substitute f(1-x) into (****):**
$$x(1 - x)\left(\frac{x(x^3 + 3x^2 + 14x - 13) - x^2(1-x)f(x)}{1-x}\right) + f(x) = x(-x^3 + 6x^2 - 23x + 5)$$
$$x(x^3 + 3x^2 + 14x - 13) - x^2(1-x)f(x) + f(x) = x(-x^3 + 6x^2 - 23x + 5)$$
$$x^4 + 3x^3 + 14x^2 - 13x - x^2(1-x)f(x) + f(x) = -x^4 + 6x^3 - 23x^2 + 5x$$
$$f(x)(1 - x^2(1-x)) = -x^4 + 6x^3 - 23x^2 + 5x - x^4 - 3x^3 - 14x^2 + 13x$$
$$f(x)(1 - x^2 + x^3) = -2x^4 + 3x^3 - 37x^2 + 18x$$
$$f(x) = \frac{-2x^4 + 3x^3 - 37x^2 + 18x}{x^3 - x^2 + 1}$$
7. **Polynomial Long Division**
Perform long division to simplify the rational function:
$$f(x) = -2x - 1 + \frac{-36x^2 + 20x +1}{x^3-x^2+1}$$
**Final Answer**
$$f(x) = -2x-1 + \frac{-36x^2 + 20x}{x^3-x^2+1}$$