Homework Help: Range of rational function

What's the definition of the range of a function?

 

Think what "domain" and "range" are. What can "x" be, and what can "x" not be? With that in mind, what can "y" be and what can "y" not be?

 

Do you know what domain and range are? Your instructor would not be giving you rational functions without a thorough treatment of the concepts of domain and range, and how they relate to rational functions.

 

[EDIT] Weird, it was all garbled on my screen when I responded, but now it looks fine, both in my quoted version and in the original post!

OK, so the function of interest is

[tex]y = \frac{1}{x+1}[/tex]

What's the domain of this function?

 

You can click on my equation (or any typeset equation on this site) and it will give you a pop-up window with the Latex code that produced it.

 

OK, good. So let's consider two cases:

[tex]x < -1[/tex]

[tex]x > -1[/tex]

Start by focusing on the first case, so we're just considering [itex]x < -1[/itex]. For [itex]x[/itex] in this range, can I make the function as big as I like? Can I make it as small as I like? If not, then what are some bounds? (Even if they're not the tightest possible bounds, it's a start.)

 

Yes, the range is expressed using y values.

What do you mean by "it would go on to infinity"? Can you make it infinitely large (positive)? Can you make it infinitely small (negative)?

If [itex]x < -1[/itex], then can y be positive at all?

 

OK, so you can make y as negative as you like by varying x over the interval [itex](-\infty, -1)[/itex]. How large can you make y if x is in the same interval?

 

No problem. It's just another way of writing

[tex]x < -1[/tex]

So how big can y get if x is in this interval? Can you make y be positive? Can you make y be zero?

 

What value of x causes y = 0?