For f(x)=2/(X-3) and g(x)=3/x, find the following composite functions and state the domain of each: ​f◦g g◦f f◦f g◦g

The required functions are f◦g = `(2x)/(3-3x)` , g◦f = `(3x-9)/2` , f◦f = `(2x - 6)/(11 -3x)` , and g◦g = x.

Expert Answers

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The function `f(x) = 2/(x - 3)` and `g(x) = 3/x.`

The composite function `f@ g(x) = f(g(x))`

`= f(3/x) = 2/((3/x-3)) `

`= 2/((3 - 3x)/x)`

`= (2x)/(3-3x)`

The composite function `g@f = g(f(x))`

`= g(2/(x - 3))`

`=3/(2/(x - 3))`

`=(3(x-3))/2`

`=(3x-9)/2`

The composite function `f@f = f(f(x))`

`=f(2/(x - 3))`

`=2/(2/(x - 3) - 3)`

`= 2/((2 -3x +9)/(x-3)`

`= (2(x-3))/(2 -3x +9)`

`= (2x - 6)/(11 -3x)`

The composite function `g@g(x) = g(g(x))`

`= g(3/x)`

`=3/(3/x)`

`=x`

For the given functions f(x)=2/(x-3) and g(x)=3/x, the required functions are f◦g = `(2x)/(3-3x)` , g◦f = `(3x-9)/2` , f◦f = `(2x - 6)/(11 -3x)` and g◦g = x.

Last Updated by eNotes Editorial on February 19, 2021
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