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

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

Solve for f (complex solution)

$$\left\{\begin{matrix}\\f=0\text{, }&\text{unconditionally}\\f\in \mathrm{C}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply f by x+y.

$$fx-fy=fx+fy$$

Subtract fx from both sides.

$$fx-fy-fx=fy$$

Combine fx and -fx to get 0.

$$-fy=fy$$

Subtract fy from both sides.

$$-fy-fy=0$$

Combine -fy and -fy to get -2fy.

$$-2fy=0$$

The equation is in standard form.

$$\left(-2y\right)f=0$$

Divide 0 by -2y.

$$f=0$$

Solve for f

$$\left\{\begin{matrix}\\f=0\text{, }&\text{unconditionally}\\f\in \mathrm{R}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply f by x+y.

$$fx-fy=fx+fy$$

Subtract fx from both sides.

$$fx-fy-fx=fy$$

Combine fx and -fx to get 0.

$$-fy=fy$$

Subtract fy from both sides.

$$-fy-fy=0$$

Combine -fy and -fy to get -2fy.

$$-2fy=0$$

The equation is in standard form.

$$\left(-2y\right)f=0$$

Divide 0 by -2y.

$$f=0$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

$$y=0\text{ or }f=0$$

Solve for x

$$x\in \mathrm{R}$$

$$y=0\text{ or }f=0$$