Solve f(x)f(2xx)=f(ax+y)

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

Solve for a

$$\left\{\begin{matrix}a=-\frac{y-2fx^{3}}{x}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&f=0\text{ or }\left(y=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply f and f to get f^{2}.

$$f^{2}x\times 2xx=f\left(ax+y\right)$$

Multiply x and x to get x^{2}.

$$f^{2}x^{2}\times 2x=f\left(ax+y\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$f^{2}x^{3}\times 2=f\left(ax+y\right)$$

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

$$f^{2}x^{3}\times 2=fax+fy$$

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

$$fax+fy=f^{2}x^{3}\times 2$$

Subtract fy from both sides.

$$fax=f^{2}x^{3}\times 2-fy$$

The equation is in standard form.

$$fxa=2f^{2}x^{3}-fy$$

Divide both sides by fx.

$$\frac{fxa}{fx}=\frac{f\left(2fx^{3}-y\right)}{fx}$$

Dividing by fx undoes the multiplication by fx.

$$a=\frac{f\left(2fx^{3}-y\right)}{fx}$$

Divide f\left(2fx^{3}-y\right) by fx.

$$a=\frac{2fx^{3}-y}{x}$$

Solve for f

$$\left\{\begin{matrix}\\f=0\text{, }&\text{unconditionally}\\f=\frac{ax+y}{2x^{3}}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$