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

Answer 1

Lincoln Anderson

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

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

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]}

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Answer 2

Lincoln Anderson

To state the domain and range of ( f(x) = \frac{x}{x - 1} ):

Domain: The domain of ( f(x) ) consists of all real numbers except ( x = 1 ), since division by zero is undefined.

Range: The range of ( f(x) ) is all real numbers except ( y = 1 ), because as ( x ) approaches 1, the function approaches positive or negative infinity.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.