Question

How do you state the domain and range of `f(x)=x/(x-1)`?

Answer 1

The domain is `x in ] -oo,1 [ uu ] 1,+oo[`

The range is `f(x) in ] -oo,1 [ uu ] 1,+oo [`

Explanation:

As you cannot divide by `0`, `x!=1`

So the domain is `x in ] -oo,1 [ uu ] 1,+oo[`

For the limits `x->+-oo`, we take the terms of highest degree in the numerator and the deniminator

`lim_(x->+-oo)f(x)=lim_(x->+-oo)x/x=1`

`lim_(x->1^(-))f(x)=1/0^(-)=-oo`

`lim_(x->1^(+))f(x)=1/0^(+)=+oo`

So the range is `f(x) in ] -oo,1 [ uu ] 1,+oo [`

graph{x/(x-1) [-10, 10, -5, 4.995]}