# The range of this function?

• Mar 17th 2010, 11:01 AM
osmosis786
The range of this function?
The function is f(x)= [6/(x+2)]-3

I wanted to find the range, also could you please post your working as I want to understand how you came to your answer, thanks.

Regards

Ossy
• Mar 17th 2010, 11:09 AM
Anonymous1
$\displaystyle f(x)= \frac{6}{x+2}-3$

Where is $\displaystyle f(x)$ undefined? Naturally we look for such things as division by $\displaystyle 0.$ When will we have division by $\displaystyle 0?$ Clearly when $\displaystyle x+2 = 0,$ $\displaystyle i.e.,$ when $\displaystyle x= -2.$ So $\displaystyle f(x)$ take on all values except when $\displaystyle x = -2.$ What is your range?
• Mar 17th 2010, 11:21 AM
skeeter
Quote:

Originally Posted by osmosis786
The function is f(x)= [6/(x+2)]-3

I wanted to find the range, also could you please post your working as I want to understand how you came to your answer, thanks.

Regards

Ossy

one method of determining the range of a function is to determine the domain of the function's inverse. of course, this depends on whether the inverse exists and is relatively easy to find.

$\displaystyle y = \frac{6}{x+2} - 3$

swap variables ...

$\displaystyle x = \frac{6}{y+2} - 3$

solve for $\displaystyle y$ ...

$\displaystyle x+3 = \frac{6}{y+2}$

$\displaystyle y+2 = \frac{6}{x+3}$

$\displaystyle y = \frac{6}{x+3} - 2$

this is the inverse function ... note that its domain is all real values $\displaystyle x$ such that $\displaystyle x \ne -3$.

so ... the range of the original function is all real $\displaystyle y$ , $\displaystyle y \ne -3$.
• Mar 17th 2010, 12:42 PM
Soroban
Hello, osmosis786!

When possible, I like to sketch a graph.

Quote:

Find the range of: .$\displaystyle f(x)\:=\:\frac{6}{x+2} -3$

The graph of .$\displaystyle y \,=\,\frac{1}{x}$ .looks like this:
Code:

                  |                   |*                   |                   | *                   |  *                   |    *                   |          *     --------------+----------------       *          |             *    |               *  |                 * |                   |                 *|                   |
Domain: .$\displaystyle x\,\neq\,0$
. Range: .$\displaystyle y\,\neq\,0$

The graph of .$\displaystyle y \:=\:\frac{1}{x+2}$
. . is the previous graph moved two units to the left.
Code:

              :  |               :*  |               :  |               : * |               :  *|               :  |*               :  |      * --------------+---+----------------   *          -2  |         *    :  |           *  :  |             * :  |               :  |             *:  |               :  |

Domain: .$\displaystyle x\,\neq\,-2$
. Range: .$\displaystyle y\,\neq\,0$

The graph of .$\displaystyle y \:=\:\frac{6}{x+2}$ . rises more "steeply"
. . and "flattens slower", but has the same basic shape.

The graph of .$\displaystyle y \:=\:\frac{6}{x+2} - 3$ . is the previous graph lowered 3 units.

Code:

              :  |               :*  |               :  |               : * | --------------+--*|--------------               :  |*               :  |      *   - - - - - - o - | - - - - - -   *        (-2,-3)|         *    :  |           *  :  |             * :  |               :  |             *:  |               :  |

Domain: .$\displaystyle x\,\neq\,-2$
. Range: .$\displaystyle y \,\neq\,-3$