Solve f^-1x=x^2+10x

$${ f }^{ -1 } x = { x }^{ 2 } +10x$$

Solve for f

$$\left\{\begin{matrix}f=\frac{1}{x+10}\text{, }&x\neq -10\\f\neq 0\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Reorder the terms.

$$\frac{1}{f}x=x^{2}+10x$$

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

$$1x=fx^{2}+10xf$$

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

$$fx^{2}+10xf=1x$$

Reorder the terms.

$$fx^{2}+10fx=x$$

Combine all terms containing f.

$$\left(x^{2}+10x\right)f=x$$

Divide both sides by x^{2}+10x.

$$\frac{\left(x^{2}+10x\right)f}{x^{2}+10x}=\frac{x}{x^{2}+10x}$$

Dividing by x^{2}+10x undoes the multiplication by x^{2}+10x.

$$f=\frac{x}{x^{2}+10x}$$

Divide x by x^{2}+10x.

$$f=\frac{1}{x+10}$$

Variable f cannot be equal to 0.

$$f=\frac{1}{x+10}\text{, }f\neq 0$$

Solve for x

$$x=-10+\frac{1}{f}$$

$$x=0\text{, }f\neq 0$$