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

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

Solve for f

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

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x+1.

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

Use the distributive property to multiply \left(-f\right)x by x+1.

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

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

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

Reorder the terms.

$$-fx^{2}-fx=x-1$$

Combine all terms containing f.

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

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

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

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

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

Divide x-1 by -x^{2}-x.

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

Solve for x (complex solution)

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

Solve for x

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