Domain and Range of a Rational Function

Question

Using Wolfram Alpha, the following rational function (x^2-1)/(x+1) has domain of {x element of R : x not=to -1} and Range {y element of R : y not=to -2}.

 

I am confused about the Range.

 

To find the Range we input {x = -1} and evaluate y, however if we do this we get an answer of ((-1)^2-1) / (-1+1) = 0 / 0 = undefined.

 

Another way to get around this is we further reduce the original funciton to (x^2-1)/(x+1) = (x-1). However, this time the Domain is R and the Range will be R too.

 

Could you please clarify this? It is frustrating not to know such a simple thing!!! 

Answer 1

Diar,

 

So, we're good on the Domain, because when you put in x = -1 into the original function, it's undefined.

 

Then, you reduced the original function to y = x - 1.  Which means that it's a line, alllllmost.  What you're describing is a "removable discontinuity" at x = -1, for exactly the reasoning you came up with.  x can't be -1 in the original function, but if you simplify the function, you end up with x+1 on both top and bottom, which cancel.

So, just to visualize what you've got, go ahead and graph the line y = x - 1.  But that's not your final graph, because there's a point on that line that isn't any good.  x can't be -1, so there's a "hole" in the graph at that value.  What is the y coordinate of the point that you just removed?

 

Does that help?

 

-- Michael