Solve f:R-Rf(x)=x+2

$$f : R - R f ( x ) = x + 2$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=\frac{R\left(x+2\right)}{1-xR^{2}}\text{, }&\left(R\neq -x^{-\frac{1}{2}}\text{ and }R\neq x^{-\frac{1}{2}}\text{ and }R\neq 0\right)\text{ or }\left(x=0\text{ and }R\neq 0\right)\\f\in \mathrm{C}\text{, }&\left(R=\frac{\sqrt{2}i}{2}\text{ or }R=-\frac{\sqrt{2}i}{2}\right)\text{ and }x=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by R.

$$f-RfxR=Rx+R\times 2$$

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

$$f-R^{2}fx=Rx+R\times 2$$

Reorder the terms.

$$-fxR^{2}+f=Rx+2R$$

Combine all terms containing f.

$$\left(-xR^{2}+1\right)f=Rx+2R$$

The equation is in standard form.

$$\left(1-xR^{2}\right)f=Rx+2R$$

Divide both sides by -xR^{2}+1.

$$\frac{\left(1-xR^{2}\right)f}{1-xR^{2}}=\frac{R\left(x+2\right)}{1-xR^{2}}$$

Dividing by -xR^{2}+1 undoes the multiplication by -xR^{2}+1.

$$f=\frac{R\left(x+2\right)}{1-xR^{2}}$$

Solve for f

$$f=\frac{R\left(x+2\right)}{1-xR^{2}}$$

$$\left(|R|\neq \frac{1}{\sqrt{x}}\text{ or }x\leq 0\right)\text{ and }R\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by R.

$$f-RfxR=Rx+R\times 2$$

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

$$f-R^{2}fx=Rx+R\times 2$$

Reorder the terms.

$$-fxR^{2}+f=Rx+2R$$

Combine all terms containing f.

$$\left(-xR^{2}+1\right)f=Rx+2R$$

The equation is in standard form.

$$\left(1-xR^{2}\right)f=Rx+2R$$

Divide both sides by -xR^{2}+1.

$$\frac{\left(1-xR^{2}\right)f}{1-xR^{2}}=\frac{R\left(x+2\right)}{1-xR^{2}}$$

Dividing by -xR^{2}+1 undoes the multiplication by -xR^{2}+1.

$$f=\frac{R\left(x+2\right)}{1-xR^{2}}$$

Solve for R (complex solution)

$$\left\{\begin{matrix}R=-\frac{\sqrt{x^{2}+4xf^{2}+4x+4}+x+2}{2fx}\text{; }R=-\frac{-\sqrt{x^{2}+4xf^{2}+4x+4}+x+2}{2fx}\text{, }&x\neq 0\text{ and }f\neq 0\\R=\frac{f}{2}\text{, }&x=0\text{ and }f\neq 0\\R\neq 0\text{, }&f=0\text{ and }x=-2\end{matrix}\right.$$

Solve for R

$$\left\{\begin{matrix}R=-\frac{\sqrt{x^{2}+4xf^{2}+4x+4}+x+2}{2fx}\text{; }R=-\frac{-\sqrt{x^{2}+4xf^{2}+4x+4}+x+2}{2fx}\text{, }&f\neq 0\text{ and }x\neq 0\text{ and }\left(|f|\leq \frac{\sqrt{-\frac{\left(x+2\right)^{2}}{x}}}{2}\text{ or }x>0\right)\\R=\frac{f}{2}\text{, }&x=0\text{ and }f\neq 0\\R\neq 0\text{, }&f=0\text{ and }x=-2\end{matrix}\right.$$