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

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

Solve for f

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

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

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+1\right)\left(x^{2}-x+1\right).

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

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

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

Use the distributive property to multiply fx^{-1}x+fx^{-1} by x^{2}-x+1 and combine like terms.

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

To multiply powers of the same base, add their exponents. Add -1 and 3 to get 2.

$$fx^{2}+fx^{-1}=1$$

Reorder the terms.

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

Multiply both sides of the equation by x.

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Reorder the terms.

$$fx^{3}+f=x$$

Combine all terms containing f.

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

Divide both sides by x^{3}+1.

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

Dividing by x^{3}+1 undoes the multiplication by x^{3}+1.

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

Divide x by x^{3}+1.

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