Let `f(x)={(x^(2)-4x+3",",x lt 3),(x-4",",x ge 3):}`and
`g(x)={(x-3",",x lt 4),(x^(2)+2x+2",",x ge 4):}`.
Describe the function `f//g` and find its domain.

Answer

`(f(x))/(g(x))={((x^(2)-4x+3)/(x-3)",",x lt 3),((x-4)/(x-3)",",3lt x lt 4),((x-4)/(x^(2)+2x+2)",",x ge 4):} `

Explanation

`f(x)={(x^(2)-4x+3",",x lt 3),(x-4",",x ge 3):}`
`={(x^(2)-4x+3",",x lt 3),(x-4",",3le x lt 4),(x-4",",x ge 4):} " (1)" `
`g(x)={(x-3",",x lt 4),(x^(2)+2x+2",",x ge 4):}`
`={(x-3",",x lt 3),(x-3",",3 le x lt 4),(x^(2)+2x+2",",x ge 4):} " (2)" `
From (1) and (2), we have
`(f(x))/(g(x))={((x^(2)-4x+3)/(x-3)",",x lt 3),((x-4)/(x-3)",",3lt x lt 4),((x-4)/(x^(2)+2x+2)",",x ge 4):} `
Clearly, `f(x)//g(x)` is not defined at `x=3`. Hence, the domain is `R-{3}.`