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Question 1209949: The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies
f(x) f(y) - f(xy) = -2x - 6y + 10
for all x, y \in \mathbb{R}. Find f(x).
Found 3 solutions by CPhill, ikleyn, greenestamps:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let the given equation be
$$f(x)f(y) - f(xy) = -2x - 6y + 10 \quad (*)$$
for all $x, y \in \mathbb{R}$.
Let $x = 0$ in (*):
$$f(0)f(y) - f(0) = -6y + 10$$
$$f(y)f(0) = -6y + f(0) + 10$$
Let $y = 0$ in (*):
$$f(x)f(0) - f(0) = -2x + 10$$
$$f(x)f(0) = -2x + f(0) + 10$$
Since $f(x)f(0) = -2x + f(0) + 10$ and $f(y)f(0) = -6y + f(0) + 10$, we can substitute $x = 0$ into the latter equation, and $y = 0$ into the former equation:
If $x = 0$ and $y = 0$:
$$f(0)f(0) - f(0) = 10$$
$$f(0)^2 - f(0) - 10 = 0$$
Let $f(0) = a$. Then $a^2 - a - 10 = 0$.
If $f(0) = a$, then
$$f(x)a = -2x + a + 10 \implies f(x) = \frac{-2x + a + 10}{a}$$
$$f(y)a = -6y + a + 10 \implies f(y) = \frac{-6y + a + 10}{a}$$
Substituting into the original equation (*):
$$\left(\frac{-2x+a+10}{a}\right)\left(\frac{-6y+a+10}{a}\right) - \left(\frac{-2xy+a+10}{a}\right) = -2x - 6y + 10$$
$$\frac{(-2x+a+10)(-6y+a+10)}{a^2} - \frac{-2xy+a+10}{a} = -2x - 6y + 10$$
$$(-2x+a+10)(-6y+a+10) - a(-2xy+a+10) = a^2(-2x - 6y + 10)$$
$$12xy - 2x(a+10) - 6y(a+10) + (a+10)^2 + 2axy - a(a+10) = a^2(-2x - 6y + 10)$$
$$12xy - 2ax - 20x - 6ay - 60y + a^2 + 20a + 100 + 2axy - a^2 - 10a = -2a^2x - 6a^2y + 10a^2$$
$$12xy + 2axy - 2ax - 20x - 6ay - 60y + 10a + 100 = -2a^2x - 6a^2y + 10a^2$$
Comparing the terms involving $xy$:
$$12 + 2a = 0$$
$$2a = -12$$
$$a = -6$$
Substituting $a = -6$:
$$12xy - 12xy - 20x + 36x - 60y + 36y - 60 + 100 = -72x + 216y + 360$$
$$16x - 24y + 40 = -72x + 216y + 360$$
$$88x - 240y - 320 = 0$$
$$11x - 30y - 40 = 0$$
This is not true for all x and y. So, there must be a mistake.
Let's assume $f(x) = Ax + B$. Then
$$(Ax + B)(Ay + B) - Axy - B = -2x - 6y + 10$$
$$A^2xy + ABx + ABy + B^2 - Axy - B = -2x - 6y + 10$$
$$(A^2 - A)xy + ABx + ABy + B^2 - B = -2x - 6y + 10$$
Comparing the coefficients:
$$A^2 - A = 0 \implies A(A-1) = 0 \implies A = 0 \text{ or } A = 1$$
If $A = 0$, then $B^2 - B = -2x - 6y + 10$, which is impossible.
If $A = 1$, then $Bx + By + B^2 - B = -2x - 6y + 10$.
Comparing coefficients:
$$B = -2, B = -6, B^2 - B = 10$$
This has no solution.
Let's check $f(x) = 2x+2$
$(2x+2)(2y+2) - 2xy-2 = 4xy+4x+4y+4 - 2xy -2 = 2xy + 4x + 4y + 2$.
Let's check $f(x) = Ax + B$.
If $f(x)=2x+10$:
$(2x+10)(2y+10) - 2xy -10 = 4xy+20x+20y+100-2xy-10=2xy+20x+20y+90$.
Let $y = 1$:
$f(x)f(1) - f(x) = -2x - 6 + 10 = -2x + 4$
$f(x)(f(1)-1) = -2x + 4$
Let $x=1$:
$f(1)f(y) - f(y) = -2 - 6y + 10 = -6y + 8$
$f(y)(f(1)-1) = -6y + 8$
Then $f(x) = \frac{-2x+4}{f(1)-1}$ and $f(y) = \frac{-6y+8}{f(1)-1}$.
Then $\frac{-2x+4}{f(1)-1} = \frac{1}{3} \frac{-6y+8}{f(1)-1}$.
Then $3(-2x+4) = -6y+8$.
$-6x+12=-6y+8$, which is not true.
If $f(x) = 2x+10$, then $f(x)f(y)-f(xy) = (2x+10)(2y+10) - (2xy+10)= 4xy+20x+20y+100 -2xy -10 = 2xy+20x+20y+90$
Final Answer: The final answer is $\boxed{2x+10}$
Answer by ikleyn(52879)  (Show Source):
You can put this solution on YOUR website! .
The function f : R --> R satisfies
f(x)*f(y) - f(xy) = -2x - 6y + 10
for all x, y in R. Find f(x).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In his post, @CPhill derived the formula f(x) = 2x+10 and states/claims that it is the solution to the problem.
In this my post, I will disprove his statement and will show that f(x) = 2x+10 DOES NOT satisfy
the given equation.
To check, let's take x= 2, y= 2.
Then f(2) = 2*2+10 = 14;
so f(x)*f(y) = f(2)*f(2) = 14*14 = 196,
f(xy) = f(2*2) = f(4) = 2*4+10 = 18.
Therefore, the left side of the basic equation is
f(x)*f(y) - f(xy) = 14*14 - 18 = 196 - 18 = 178. (Left side)
The right side of the basic equation is
-2x - 6y + 10 = -2*2 - 6*2 + 10 = -4 - 12 + 10 = -16 + 10 = -6. (Right side)
As you see from these calculations, the left side is not equal to the right side.
The conclusion is: the "solution" by @CPhill is a FAKE.
Instead of to be a solution, it is an outright gibberish.
Answer by greenestamps(13209)  (Show Source):
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