Solve f^-1=-2x+1/x+4

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

Solve for f

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

$$x\neq -4\text{ and }x\neq \frac{1}{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+4.

$$\left(x+4\right)f^{-1}=-2x+1$$

Use the distributive property to multiply x+4 by f^{-1}.

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

Reorder the terms.

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

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

$$1x+4\times 1=-2xf+f$$

Multiply 4 and 1 to get 4.

$$1x+4=-2xf+f$$

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

$$-2xf+f=1x+4$$

Reorder the terms.

$$-2fx+f=x+4$$

Combine all terms containing f.

$$\left(-2x+1\right)f=x+4$$

The equation is in standard form.

$$\left(1-2x\right)f=x+4$$

Divide both sides by -2x+1.

$$\frac{\left(1-2x\right)f}{1-2x}=\frac{x+4}{1-2x}$$

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

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

Variable f cannot be equal to 0.

$$f=\frac{x+4}{1-2x}\text{, }f\neq 0$$

Solve for x

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

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

Steps for Solving Linear Equation

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$\left(x+4\right)f^{-1}=-2x+1$$

Use the distributive property to multiply x+4 by f^{-1}.

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

Add 2x to both sides.

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

Subtract 4f^{-1} from both sides.

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

Reorder the terms.

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

Multiply both sides of the equation by f.

$$2xf+1x=f-4$$

Reorder the terms.

$$2fx+x=f-4$$

Combine all terms containing x.

$$\left(2f+1\right)x=f-4$$

Divide both sides by 1+2f.

$$\frac{\left(2f+1\right)x}{2f+1}=\frac{f-4}{2f+1}$$

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

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

Variable x cannot be equal to -4.

$$x=\frac{f-4}{2f+1}\text{, }x\neq -4$$