Solve f(x+y)+f(x-y)=2f(x)+2f(y)

$$f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y )$$

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.

$$fx+fy+f\left(x-y\right)=2fx+2fy$$

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

$$fx+fy+fx-fy=2fx+2fy$$

Combine fx and fx to get 2fx.

$$2fx+fy-fy=2fx+2fy$$

Combine fy and -fy to get 0.

$$2fx=2fx+2fy$$

Subtract 2fx from both sides.

$$2fx-2fx=2fy$$

Combine 2fx and -2fx to get 0.

$$0=2fy$$

Swap sides so that all variable terms are on the left hand side.

$$2fy=0$$

The equation is in standard form.

$$2yf=0$$

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.

$$fx+fy+f\left(x-y\right)=2fx+2fy$$

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

$$fx+fy+fx-fy=2fx+2fy$$

Combine fx and fx to get 2fx.

$$2fx+fy-fy=2fx+2fy$$

Combine fy and -fy to get 0.

$$2fx=2fx+2fy$$

Subtract 2fx from both sides.

$$2fx-2fx=2fy$$

Combine 2fx and -2fx to get 0.

$$0=2fy$$

Swap sides so that all variable terms are on the left hand side.

$$2fy=0$$

The equation is in standard form.

$$2yf=0$$

Divide 0 by 2y.

$$f=0$$

Solve for x (complex solution)

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

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

Solve for x

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

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