Solve 1/xf(-x)+f(1/x)=x

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

Solve for f

$$f=\frac{x^{2}}{1-x}$$

$$x\neq 0\text{ and }x\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

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

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

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

Reorder the terms.

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

Reorder the terms.

$$-fx+f=x^{2}$$

Combine all terms containing f.

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

The equation is in standard form.

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

Divide both sides by -x+1.

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

Dividing by -x+1 undoes the multiplication by -x+1.

$$f=\frac{x^{2}}{1-x}$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{f\left(f+4\right)}-f}{2}$$

$$x=\frac{\sqrt{f\left(f+4\right)}-f}{2}\text{, }f\neq 0$$

Solve for x

$$x=\frac{-\sqrt{f\left(f+4\right)}-f}{2}$$

$$x=\frac{\sqrt{f\left(f+4\right)}-f}{2}\text{, }f\leq -4\text{ or }f>0$$