Question

The function \(f : \mathbb{R} \rightarrow \mathbb{R}\) satisfies \(f(x) f(y) - f(xy) = x + y\) for all \(x\), \(y \in \mathbb{R}\). Find \(f(x)\).

Answer 1

let \(y = 0\),
\[f(x)f(0)-f(0) = x\tag{1}\]

let \(x=0\) in above equation :
\[f(0)f(0)-f(0) = 0\implies f(0) = 0\lor 1\]
however \(f(0)=0\) is not possible (why?)
therefore \(f(0)=1\)

plug that back in equation \((1)\) to get \(f(x) = x+1\)

Answer 2

I was just fixing to ask why :p

Answer 3

I also noticed I get two possibilities for f(1)

Answer 4

I don't know how to know which one is the right one for f(1)

Answer 5

\[
\text{Assume }f(0)=0\\
\begin{align*}
f(0)f(x)-f(0)&=x\\
0&=x
\end{align*}
\]

Answer 6

\[f^2(1)-f(1)-2=0 \\ (f(1)-2)(f(1)+1)=0 \\ f(1)=2 \text{ or } f(1)=-1\]

Answer 7

I also did a similar thing then ganeshie8 posted his answer.

Answer 8

just an example to convince myself on why there are false values :
\(f = -1 \implies f^2 = 1 \implies f = \pm 1 \)

ofcourse \(f=1\) is extraneous.
since we're squaring the function, we are increasing the degree and thus increasing the number of solutions, so we must check for extraneous stuff in the end..

Answer 9

\[
\text{Assume }f(1)=-1\\
\begin{align*}
f(1)f(x)-f(x)&=x+1\\
-2f(x)&=x+1\\
f(x)&=-\frac{x+1}{2}\\
f(x)f(y)-f(xy)&=\frac{x+1}{2}\frac{y+1}{2}-\frac{xy+1}{2}\\
&=\frac{xy+x+y+1}{4}-\frac{xy+1}{2}\\
&\neq x+y
\end{align*}
\]

Answer 10

ok cool and if f(1)=2
\[f(1)f(x)-f(x)=x+1 \\ 2f(x)-f(x)=x+1 \\ f(x)=x+1 \\ \text{ checking } \\ f(x)f(y)-f(xy)=(x+1)(y+1)-(xy+1) \\ = xy+x+y+1-xy-1 \\ =x+y\]
great stuff @thomas5267

Answer 11

One of the question on the only mathematics competition I have participated is to find the value of \(f(0)\) with \(f(x)f(y)=f(x+y)\). It can be solved with a similar method.

Answer 12

@zmudz can you give us some background on these problems? Are you looking for hints, solution methods or the solution?

Answer 13

i assumed its one to one and onto function :3


f(x) f(y) - f(xy) = x + y

yx-xy=f(y)+f(x)
xy-xy=f(x)+f(y)
0=f(x)+f(y)

f(x)+f(y)=0
f(x)=-f(y)

so f(x)=-x satisfy :)

Answer 14

I can't figure out how you got any of that.
But checking your solution...
we see that
\[-x \cdot (-y)-(-xy) \neq x+y\]

Answer 15

hmmm you are right