# Thread: Confusing range of a Relation.

1. ## Confusing range of a Relation.

Okay. So our assignment is to get the domain, range, intercepts, etc. of the relation (also a function but not expressed as one)
$\displaystyle 5y(x-3)(x-1)=2(5x+3)$. I didn't have a problem at all till I realize that I got $\displaystyle (-\infty,-5]\cup[-1/5,0)\cup(0,+\infty)$
(I EXCLUDED ZERO because when I solved for x, y is in the DENOMINATOR) as the range and have an x-intercept $\displaystyle (-3/5,0)$
which is a contradiction with the range.

Now, it leaves me with a dilemma because I can't include zero in the range because x will be undefined (at least if you use quadratic formula by manipulating the given equation which I think alters its nature I presume), and I can't just simply remove the x-intercept because of the same reason.

Here's my question, do I include zero in the range OR remove the x-intercept OR just leave my answer as it is.

2. ## Re: Confusing range of a Relation.

It seems to me that $\displaystyle y \in \mathbb{R}$ and $\displaystyle x \in (-\infty, 1) \cup ( 1,3) \cup (3,\infty)$ , your x-intercept is correct. Include y=0 in the range as it gives a corresponding x-value.

What about when $\displaystyle x=0$ ?

3. ## Re: Confusing range of a Relation.

Originally Posted by pickslides
It seems to me that $\displaystyle y \in \mathbb{R}$
Huh? Do you mean that the range is all real numbers? I think you must recheck it (or maybe I'm the one who is wrong).

4. ## Re: Confusing range of a Relation.

Hello, Kaloda!

I think you're confusing yourself . . .

Find the domain, range, intercepts, etc. of the relation: .$\displaystyle 5y(x-3)(x-1)\:=\:2(5x+3)$

We have: .$\displaystyle y \;=\;\frac{2}{5}\cdot\frac{5x+3}{(x-3)(x-1)}$

If $\displaystyle y = 0\!:\;x = \text{-}\tfrac{3}{5}\;\hdots\;x\text{-intercept: }\left(\text{-}\tfrac{3}{5},\:0\right)$

If $\displaystyle x = 0\!:\;y = \tfrac{2}{5}\;\hdots\;y\text{-intercept: }\left(0,\:\tfrac{2}{5}\right)$

Horizontal asymptote: .$\displaystyle y = 0$
Vertical asymptotes: .$\displaystyle x = 1,\;x = 3$

Domain: .$\displaystyle x \ne 1,\:x \ne 3$
. . . . . . . $\displaystyle (\text{-}\infty,\,1)\,\cup\,(1,\,3)\,\cup\,(3,\,\infty)$

Range: difficult to determine without using Calculus.

The graph looks something like this:

Code:
                        |
| :         :
| :         :*
|*:         :
| :         : *
| :         :  *
oB:         :     *
A | :         :          *
- -------------------o+-:---------:--------------
*             * | :1        :3
*     *   | :         :
o      | :         :
D      | :    C    :
| :    o    :
| :  *   *  :
| : *     * :
| :         :
| :*        *:
$\displaystyle A\text{ and }B\text{ are the }x\text{- and }y\text{-intercepts, respectively.}$

$\displaystyle C\text{ has coordinates: }\,\left(\tfrac{9}{5},\text{-}5\right)$
$\displaystyle D\text{ has coordinates: }\,\left(\text{-}3,\,\text{-}\tfrac{1}{5}\right)$

Therefore, the range is: .$\displaystyle \left(\text{-}\infty,\,\text{-}5)\,\cup\,\leeft(\text{-}\tfrac{1}{5},\,\infty\right)$