Solve g^-1(x)=4/3-x

$$g ^ { - 1 } ( x ) = \frac { 4 } { 3 - x }$$

Solve for g

$$g=\frac{x\left(3-x\right)}{4}$$

$$x\neq 3\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by -x+3.

$$g^{-1}x\left(-x+3\right)=4$$

Use the distributive property to multiply g^{-1}x by -x+3.

$$-g^{-1}x^{2}+3g^{-1}x=4$$

Reorder the terms.

$$-\frac{1}{g}x^{2}+3\times \frac{1}{g}x=4$$

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

$$-x^{2}+3\times 1x=4g$$

Do the multiplications.

$$-x^{2}+3x=4g$$

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

$$4g=-x^{2}+3x$$

The equation is in standard form.

$$4g=3x-x^{2}$$

Divide both sides by 4.

$$\frac{4g}{4}=\frac{x\left(3-x\right)}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$g=\frac{x\left(3-x\right)}{4}$$

Variable g cannot be equal to 0.

$$g=\frac{x\left(3-x\right)}{4}\text{, }g\neq 0$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{9-16g}+3}{2}$$

$$x=\frac{\sqrt{9-16g}+3}{2}\text{, }g\neq 0$$

Solve for x

$$x=\frac{-\sqrt{9-16g}+3}{2}$$

$$x=\frac{\sqrt{9-16g}+3}{2}\text{, }g\neq 0\text{ and }g\leq \frac{9}{16}$$