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

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

Solve for f

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

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

Steps for Solving Linear Equation

Reorder the terms.

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

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2f, the least common multiple of f,2.

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

Multiply 2 and 1 to get 2.

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

Multiply \frac{1}{2} and 2 to get 1.

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

Multiply 2 and -1 to get -2.

$$2x=xf-2f$$

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

$$xf-2f=2x$$

Combine all terms containing f.

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

Divide both sides by x-2.

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

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

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

Variable f cannot be equal to 0.

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

Solve for x

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

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

Steps for Solving Linear Equation

Subtract \frac{1}{2}x from both sides.

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

Reorder the terms.

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

Multiply both sides of the equation by 2f, the least common multiple of 2,f.

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

Multiply -\frac{1}{2} and 2 to get -1.

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

Multiply 2 and 1 to get 2.

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

Combine all terms containing x.

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

The equation is in standard form.

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

Divide both sides by 2-f.

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

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

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