Question

How do you find domain and range for `f(x)= x/(x-2)`?

Answer 1

Domain: `RR - {2}`
Range: `RR - {1}`

Explanation:

`f(x)=2/x-2` is clearly defined for all Real values of `x` except `x=2` (since division by `0` is undefined).
Therefore the Domain is all Real values except `2`

To determine the range, consider the possible limitations on `g(x)` where `g(x)` is the inverse of `f(x)`

By definition of inverse
`color(white)("XXX")f(g(x)) = x`
and from the given definition of `f(x)`
`color(white)("XXX")f(g(x)) = g(x)/(g(x)-2)`

Therefore
`color(white)("XXX")g(x)/(g(x)-2) = x`

`color(white)("XXX")g(x) = x*g(x) -x*2`

`color(white)("XXX")g(x)-x*g(x) = -2x`

`color(white)("XXX")g(x)(1-x) = -2x`

`color(white)("XXX")g(x) = (-2x)/(1-x)`

which is defined for all values of `x!=1`

Therefore the Range is all Real values except `1`