Solve f(x)-2f(1/x)=2x-1

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

Solve for f (complex solution)

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

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$fxx-2f\times 1=2xx+x\left(-1\right)$$

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

$$fx^{2}-2f\times 1=2xx+x\left(-1\right)$$

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

$$fx^{2}-2f\times 1=2x^{2}+x\left(-1\right)$$

Multiply 2 and 1 to get 2.

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

Combine all terms containing f.

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

The equation is in standard form.

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

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

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

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

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

Solve for f

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

$$|x|\neq \sqrt{2}\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$fxx-2f\times 1=2xx+x\left(-1\right)$$

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

$$fx^{2}-2f\times 1=2xx+x\left(-1\right)$$

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

$$fx^{2}-2f\times 1=2x^{2}+x\left(-1\right)$$

Multiply 2 and 1 to get 2.

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

Combine all terms containing f.

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

The equation is in standard form.

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

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

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

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{8f^{2}-16f+1}+1}{2\left(f-2\right)}\text{, }&f\neq 2\\x=\frac{\sqrt{8f^{2}-16f+1}-1}{2\left(f-2\right)}\text{, }&f\neq 0\text{ and }f\neq 2\\x=4\text{, }&f=2\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{8f^{2}-16f+1}+1}{2\left(f-2\right)}\text{, }&f\leq -\frac{\sqrt{14}}{4}+1\text{ or }\left(f\neq 2\text{ and }f\geq \frac{\sqrt{14}}{4}+1\right)\\x=\frac{\sqrt{8f^{2}-16f+1}-1}{2\left(f-2\right)}\text{, }&\left(f\neq 0\text{ and }f\leq -\frac{\sqrt{14}}{4}+1\right)\text{ or }\left(f\neq 2\text{ and }f\geq \frac{\sqrt{14}}{4}+1\right)\\x=4\text{, }&f=2\end{matrix}\right.$$