Solve f^-1(x)=3x

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=0\text{, }&f\neq 0\\x\in \mathrm{C}\text{, }&f=\frac{1}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 3x from both sides.

$$f^{-1}x-3x=0$$

Reorder the terms.

$$-3x+\frac{1}{f}x=0$$

Multiply both sides of the equation by f.

$$-3xf+1x=0$$

Reorder the terms.

$$-3fx+x=0$$

Combine all terms containing x.

$$\left(-3f+1\right)x=0$$

The equation is in standard form.

$$\left(1-3f\right)x=0$$

Divide 0 by 1-3f.

$$x=0$$

Solve for f

$$\left\{\begin{matrix}\\f=\frac{1}{3}\approx 0.333333333\text{, }&\text{unconditionally}\\f\neq 0\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Reorder the terms.

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

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

$$1x=3xf$$

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

$$3xf=1x$$

Reorder the terms.

$$3fx=x$$

The equation is in standard form.

$$3xf=x$$

Divide both sides by 3x.

$$\frac{3xf}{3x}=\frac{x}{3x}$$

Dividing by 3x undoes the multiplication by 3x.

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

Divide x by 3x.

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

Solve for x

$$\left\{\begin{matrix}x=0\text{, }&f\neq 0\\x\in \mathrm{R}\text{, }&f=\frac{1}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 3x from both sides.

$$f^{-1}x-3x=0$$

Reorder the terms.

$$-3x+\frac{1}{f}x=0$$

Multiply both sides of the equation by f.

$$-3xf+1x=0$$

Reorder the terms.

$$-3fx+x=0$$

Combine all terms containing x.

$$\left(-3f+1\right)x=0$$

The equation is in standard form.

$$\left(1-3f\right)x=0$$

Divide 0 by 1-3f.

$$x=0$$