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

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

f uchun yechish (complex solution)

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

Chiziqli tenglamalarni yechish bo'yicha qadamlar

f ga x+y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

f ga x-y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

2fx ni olish uchun fx va fx ni birlashtirish.

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

0 ni olish uchun fy va -fy ni birlashtirish.

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

Ikkala tarafdan 2fx ni ayirish.

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

0 ni olish uchun 2fx va -2fx ni birlashtirish.

$$0=2fy$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$2fy=0$$

Tenglama standart shaklda.

$$2yf=0$$

0 ni 2y ga bo'lish.

$$f=0$$

f uchun yechish

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

Chiziqli tenglamalarni yechish bo'yicha qadamlar

f ga x+y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

f ga x-y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

2fx ni olish uchun fx va fx ni birlashtirish.

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

0 ni olish uchun fy va -fy ni birlashtirish.

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

Ikkala tarafdan 2fx ni ayirish.

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

0 ni olish uchun 2fx va -2fx ni birlashtirish.

$$0=2fy$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$2fy=0$$

Tenglama standart shaklda.

$$2yf=0$$

0 ni 2y ga bo'lish.

$$f=0$$

x uchun yechish (complex solution)

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

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

x uchun yechish

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

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