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.

$$gx=gx$$

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.

$$fx+gx=fx+gx$$

Subtract gx from 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.

$$gx=gx$$

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.

$$fx+gx=fx+gx$$

Subtract gx from 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.

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