Solve f(x)+2f(1/x)=3x+2

$$f ( x ) + 2 f ( \frac { 1 } { x } ) = 3 x + 2$$

Solve for f (complex solution)

$$f=\frac{x\left(3x+2\right)}{x^{2}+2}$$

$$x\neq -\sqrt{2}i\text{ and }x\neq \sqrt{2}i\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$fxx+2f\times 1=3xx+x\times 2$$

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

$$fx^{2}+2f\times 1=3xx+x\times 2$$

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

$$fx^{2}+2f\times 1=3x^{2}+x\times 2$$

Multiply 2 and 1 to get 2.

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

Combine all terms containing f.

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

The equation is in standard form.

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

Divide both sides by x^{2}+2.

$$\frac{\left(x^{2}+2\right)f}{x^{2}+2}=\frac{x\left(3x+2\right)}{x^{2}+2}$$

Dividing by x^{2}+2 undoes the multiplication by x^{2}+2.

$$f=\frac{x\left(3x+2\right)}{x^{2}+2}$$

Solve for f

$$f=\frac{x\left(3x+2\right)}{x^{2}+2}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$fxx+2f\times 1=3xx+x\times 2$$

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

$$fx^{2}+2f\times 1=3xx+x\times 2$$

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

$$fx^{2}+2f\times 1=3x^{2}+x\times 2$$

Multiply 2 and 1 to get 2.

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

Combine all terms containing f.

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

The equation is in standard form.

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

Divide both sides by x^{2}+2.

$$\frac{\left(x^{2}+2\right)f}{x^{2}+2}=\frac{x\left(3x+2\right)}{x^{2}+2}$$

Dividing by x^{2}+2 undoes the multiplication by x^{2}+2.

$$f=\frac{x\left(3x+2\right)}{x^{2}+2}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{1+6f-2f^{2}}+1}{f-3}\text{, }&f\neq 0\text{ and }f\neq 3\\x=\frac{\sqrt{1+6f-2f^{2}}+1}{f-3}\text{, }&f\neq 3\\x=3\text{, }&f=3\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{1+6f-2f^{2}}+1}{f-3}\text{, }&f\neq 3\text{ and }f\neq 0\text{ and }f\geq \frac{3-\sqrt{11}}{2}\text{ and }f\leq \frac{\sqrt{11}+3}{2}\\x=\frac{\sqrt{1+6f-2f^{2}}+1}{f-3}\text{, }&f\neq 3\text{ and }f\geq \frac{3-\sqrt{11}}{2}\text{ and }f\leq \frac{\sqrt{11}+3}{2}\\x=3\text{, }&f=3\end{matrix}\right.$$