Solve f^-1(x)=2x+1/y

$$f ^ { - 1 } ( x ) = \frac { 2 x + 1 } { y }$$

Solve for f

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

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

Steps for Solving Linear Equation

Multiply both sides of the equation by y.

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

Reorder the terms.

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

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

$$1xy=2xf+f$$

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

$$2xf+f=1xy$$

Reorder the terms.

$$2fx+f=xy$$

Combine all terms containing f.

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

Divide both sides by 2x+1.

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

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

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

Variable f cannot be equal to 0.

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

Solve for x

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

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

Steps for Solving Linear Equation

Multiply both sides of the equation by y.

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

Subtract 2x from both sides.

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

Reorder the terms.

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

Multiply both sides of the equation by f.

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

Reorder the terms.

$$xy-2fx=f$$

Combine all terms containing x.

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

Divide both sides by y-2f.

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

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

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