Solve f^-1(x)=3x-x^2

$$f ^ { - 1 } ( x ) = 3 x - x ^ { 2 }$$

Solve for f

$$\left\{\begin{matrix}f=-\frac{1}{x-3}\text{, }&x\neq 3\\f\neq 0\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Reorder the terms.

$$\frac{1}{f}x=-x^{2}+3x$$

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

$$1x=-x^{2}f+3xf$$

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

$$-x^{2}f+3xf=1x$$

Reorder the terms.

$$-fx^{2}+3fx=x$$

Combine all terms containing f.

$$\left(-x^{2}+3x\right)f=x$$

The equation is in standard form.

$$\left(3x-x^{2}\right)f=x$$

Divide both sides by 3x-x^{2}.

$$\frac{\left(3x-x^{2}\right)f}{3x-x^{2}}=\frac{x}{3x-x^{2}}$$

Dividing by 3x-x^{2} undoes the multiplication by 3x-x^{2}.

$$f=\frac{x}{3x-x^{2}}$$

Divide x by 3x-x^{2}.

$$f=\frac{1}{3-x}$$

Variable f cannot be equal to 0.

$$f=\frac{1}{3-x}\text{, }f\neq 0$$

Solve for x

$$x=3-\frac{1}{f}$$

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