Question

F(x+1)+f(x+2)=2x+3; f(x)=?

Answer 1

`F=(-xf+f+2x)/(x+1);\quad \:x\\e \:-1`

Explanation:

`1. \mathrm{Subtract\:}f\left(x+2\right)\mathrm{\:from\:both\:sides}\\ F\left(x+1\right)+f\left(x+2\right)-f\left(x+2\right)=2x+3f-f\left(x+2\right)`

`2. \mathrm{Simplify}\\ F\left(x+1\right)=-xf+f+2x\\`

`3. \mathrm{Divide\:both\:sides\:by\:}x+1;\quad \:x\\e \:-1\\ (F\left(x+1\right))/(x+1)=-(xf)/(x+1)+(f)/(x+1)+(2x)/(x+1);\quad \:x\\e \:-1`

`4. \mathrm{Simplify}\\ F=(-xf+f+2x)/(x+1);\quad \:x\\e \:-1`

Final Answer:
`F=(-xf+f+2x)/(x+1);\quad \:x\\e \:-1`