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

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

Solve for f (complex solution)

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

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

Steps for Solving Linear Equation

Reorder the terms.

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

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by f.

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

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

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

Reorder the terms.

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

Combine all terms containing f.

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

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

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

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

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

Variable f cannot be equal to 0.

$$f=\frac{x}{x^{2}-2}\text{, }f\neq 0$$

Solve for f

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

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

Steps for Solving Linear Equation

Reorder the terms.

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

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by f.

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

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

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

Reorder the terms.

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

Combine all terms containing f.

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

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

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

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

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

Variable f cannot be equal to 0.

$$f=\frac{x}{x^{2}-2}\text{, }f\neq 0$$

Solve for x

$$x=-\frac{\sqrt{8f^{2}+1}-1}{2f}$$

$$x=\frac{\sqrt{8f^{2}+1}+1}{2f}\text{, }f\neq 0$$