Solve x^3=f(-x)

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

Solve for f (complex solution)

$$\left\{\begin{matrix}\\f=-x^{2}\text{, }&\text{unconditionally}\\f\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Reorder the terms.

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

The equation is in standard form.

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

Divide both sides by -x.

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

Dividing by -x undoes the multiplication by -x.

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

Divide x^{3} by -x.

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

Solve for f

$$\left\{\begin{matrix}\\f=-x^{2}\text{, }&\text{unconditionally}\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Reorder the terms.

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

The equation is in standard form.

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

Divide both sides by -x.

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

Dividing by -x undoes the multiplication by -x.

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

Divide x^{3} by -x.

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

Solve for x (complex solution)

$$x=i\sqrt{f}$$

$$x=0$$

$$x=-i\sqrt{f}$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\sqrt{-f}\text{; }x=-\sqrt{-f}\text{, }&f\leq 0\end{matrix}\right.$$