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

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

Solve for f

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

$$x\neq 0\text{ and }x\neq 3$$

Steps for Solving Linear Equation

Reorder the terms.

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

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

$$1x=f\times 3-xf$$

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

$$f\times 3-xf=1x$$

Reorder the terms.

$$-fx+3f=x$$

Combine all terms containing f.

$$\left(-x+3\right)f=x$$

The equation is in standard form.

$$\left(3-x\right)f=x$$

Divide both sides by -x+3.

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

Dividing by -x+3 undoes the multiplication by -x+3.

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

Variable f cannot be equal to 0.

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

Solve for x

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

$$f\neq -1\text{ and }f\neq 0$$

Steps for Solving Linear Equation

Add x to both sides.

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

Reorder the terms.

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

Multiply both sides of the equation by f.

$$fx+1x=3f$$

Reorder the terms.

$$fx+x=3f$$

Combine all terms containing x.

$$\left(f+1\right)x=3f$$

Divide both sides by 1+f.

$$\frac{\left(f+1\right)x}{f+1}=\frac{3f}{f+1}$$

Dividing by 1+f undoes the multiplication by 1+f.

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