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

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

Solve for f (complex solution)

$$f\in \mathrm{C}$$

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

Solution Steps

Multiply both sides of the equation by gx.

$$fx=fx\left(gx\right)^{-1}gx$$

Multiply x and x to get x^{2}.

$$fx=fx^{2}\left(gx\right)^{-1}g$$

Expand \left(gx\right)^{-1}.

$$fx=fx^{2}g^{-1}x^{-1}g$$

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

$$fx=fx^{1}g^{-1}g$$

Calculate x to the power of 1 and get x.

$$fx=fxg^{-1}g$$

Subtract fxg^{-1}g from both sides.

$$fx-fxg^{-1}g=0$$

Reorder the terms.

$$fx-\frac{1}{g}fgx=0$$

Multiply both sides of the equation by g.

$$fxg-\frac{1}{g}fgxg=0$$

Multiply g and g to get g^{2}.

$$fxg-\frac{1}{g}fg^{2}x=0$$

Express \frac{1}{g}f as a single fraction.

$$fxg-\frac{f}{g}g^{2}x=0$$

Express \frac{f}{g}g^{2} as a single fraction.

$$fxg-\frac{fg^{2}}{g}x=0$$

Cancel out g in both numerator and denominator.

$$fxg-fgx=0$$

Combine fxg and -fgx to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any f.

$$f\in \mathrm{C}$$

Solve for f

$$f\in \mathrm{R}$$

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

Solution Steps

Multiply both sides of the equation by gx.

$$fx=fx\left(gx\right)^{-1}gx$$

Multiply x and x to get x^{2}.

$$fx=fx^{2}\left(gx\right)^{-1}g$$

Expand \left(gx\right)^{-1}.

$$fx=fx^{2}g^{-1}x^{-1}g$$

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

$$fx=fx^{1}g^{-1}g$$

Calculate x to the power of 1 and get x.

$$fx=fxg^{-1}g$$

Subtract fxg^{-1}g from both sides.

$$fx-fxg^{-1}g=0$$

Reorder the terms.

$$fx-\frac{1}{g}fgx=0$$

Multiply both sides of the equation by g.

$$fxg-\frac{1}{g}fgxg=0$$

Multiply g and g to get g^{2}.

$$fxg-\frac{1}{g}fg^{2}x=0$$

Express \frac{1}{g}f as a single fraction.

$$fxg-\frac{f}{g}g^{2}x=0$$

Express \frac{f}{g}g^{2} as a single fraction.

$$fxg-\frac{fg^{2}}{g}x=0$$

Cancel out g in both numerator and denominator.

$$fxg-fgx=0$$

Combine fxg and -fgx to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any f.

$$f\in \mathrm{R}$$

Solve for g

$$g\neq 0$$

$$x\neq 0$$

Solution Steps

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

$$fx=fx\left(gx\right)^{-1}gx$$

Multiply x and x to get x^{2}.

$$fx=fx^{2}\left(gx\right)^{-1}g$$

Expand \left(gx\right)^{-1}.

$$fx=fx^{2}g^{-1}x^{-1}g$$

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

$$fx=fx^{1}g^{-1}g$$

Calculate x to the power of 1 and get x.

$$fx=fxg^{-1}g$$

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

$$fxg^{-1}g=fx$$

Reorder the terms.

$$\frac{1}{g}fgx=fx$$

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

$$1fgx=fxg$$

Subtract fxg from both sides.

$$1fgx-fxg=0$$

Combine 1fgx and -fxg to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any g.

$$g\in \mathrm{R}$$

Variable g cannot be equal to 0.

$$g\in \mathrm{R}\setminus 0$$