If `f,g: RvecR` are defined respectively by `f(x)=x^2+3x+1,g(x)=2x-3,` find fog (ii) gof (iii) fof (iv) gog.

Answer

N/a

Explanation

Given that, `f(x)=x^(2)+3x+1, g(x)=2x-3`
(i) `fog=f{g(x)}=f(2x-3)`
`=(2x-3)^(2) +3(2x-3) +1`
`=4x^(2) +9-12x+6x-9+1 =4x^(2)-6x+1`
(ii) `gof=g{f(x)}=g(x^(2)+3x+1)`
`=2(x^(2)+3x+1)-3`
`=2x^(2)+6x+2-3=2x^(2)+6x-1`
(iii) `fof=f{f(x)}=f(x^(2)+3x+1)`
`=(x^(2)+3x+1)^(2) +3(x^(2)+3x+1)+1`
`=x^(4)+9x^(2)+1+6x^(3)+6x+2x^(2)+3x^(2)+9x+3+1`
`= x^(4)+6x^(3)+14x^(2)+15x+5`
(iv) ` gog =g{g(x)}=g(2x-3)`
`implies 2(2x-3)-3`
`=4x-6-3=4x-9`