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

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

Solve for f

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

$$x\neq 0\text{ and }x\neq \frac{1}{2}$$

Steps for Solving Linear Equation

Reorder the terms.

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

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

$$1x=2xf+f\left(-1\right)$$

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

$$2xf+f\left(-1\right)=1x$$

Reorder the terms.

$$2fx-f=x$$

Combine all terms containing f.

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

Divide both sides by 2x-1.

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

Dividing by 2x-1 undoes the multiplication by 2x-1.

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

Variable f cannot be equal to 0.

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

Solve for x

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

$$f\neq \frac{1}{2}\text{ and }f\neq 0$$

Steps for Solving Linear Equation

Subtract 2x from both sides.

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

Reorder the terms.

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

Multiply both sides of the equation by f.

$$-2xf+1x=-f$$

Reorder the terms.

$$-2fx+x=-f$$

Combine all terms containing x.

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

The equation is in standard form.

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

Divide both sides by 1-2f.

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

Dividing by 1-2f undoes the multiplication by 1-2f.

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