Solve (f-g)(x)=f(x)-g(x)

$$( f - g ) ( x ) = f ( x ) - g ( x )$$

Solve for f (complex solution)

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

Solution Steps

Use the distributive property to multiply f-g by x.

$$fx-gx=fx-gx$$

Subtract fx from both sides.

$$fx-gx-fx=-gx$$

Combine fx and -fx to get 0.

$$-gx=-gx$$

Cancel out -1 on both sides.

$$gx=gx$$

Reorder the terms.

$$\text{true}$$

This is true for any f.

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

Solve for g (complex solution)

$$g\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply f-g by x.

$$fx-gx=fx-gx$$

Add gx to both sides.

$$fx-gx+gx=fx$$

Combine -gx and gx to get 0.

$$fx=fx$$

Reorder the terms.

$$\text{true}$$

This is true for any g.

$$g\in \mathrm{C}$$

Solve for f

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

Solution Steps

Use the distributive property to multiply f-g by x.

$$fx-gx=fx-gx$$

Subtract fx from both sides.

$$fx-gx-fx=-gx$$

Combine fx and -fx to get 0.

$$-gx=-gx$$

Cancel out -1 on both sides.

$$gx=gx$$

Reorder the terms.

$$\text{true}$$

This is true for any f.

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

Solve for g

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

Solution Steps

Use the distributive property to multiply f-g by x.

$$fx-gx=fx-gx$$

Add gx to both sides.

$$fx-gx+gx=fx$$

Combine -gx and gx to get 0.

$$fx=fx$$

Reorder the terms.

$$\text{true}$$

This is true for any g.

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