Solve f^-1(x)=4/x-1

$$f ^ { - 1 } ( x ) = \frac { 4 } { x - 1 }$$

Solve for f

$$f=\frac{x\left(x-1\right)}{4}$$

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x-1.

$$f^{-1}x\left(x-1\right)=4$$

Use the distributive property to multiply f^{-1}x by x-1.

$$f^{-1}x^{2}-f^{-1}x=4$$

Reorder the terms.

$$\frac{1}{f}x^{2}-\frac{1}{f}x=4$$

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

$$1x^{2}-x=4f$$

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

$$4f=1x^{2}-x$$

Reorder the terms.

$$4f=x^{2}-x$$

Divide both sides by 4.

$$\frac{4f}{4}=\frac{x\left(x-1\right)}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$f=\frac{x\left(x-1\right)}{4}$$

Variable f cannot be equal to 0.

$$f=\frac{x\left(x-1\right)}{4}\text{, }f\neq 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{16f+1}+1}{2}$$

$$x=\frac{-\sqrt{16f+1}+1}{2}\text{, }f\neq 0$$

Solve for x

$$x=\frac{\sqrt{16f+1}+1}{2}$$

$$x=\frac{-\sqrt{16f+1}+1}{2}\text{, }f\geq -\frac{1}{16}\text{ and }f\neq 0$$