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

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

Solve for f

$$f=x$$

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x+1.

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

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

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

Reorder the terms.

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

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

$$1x^{2}+1x=fx+f$$

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

$$fx+f=1x^{2}+1x$$

Reorder the terms.

$$fx+f=x^{2}+x$$

Combine all terms containing f.

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

Divide both sides by x+1.

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

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

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

Divide x\left(1+x\right) by x+1.

$$f=x$$

Variable f cannot be equal to 0.

$$f=x\text{, }f\neq 0$$

Solve for x

$$x=f$$

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