# Thread: The range of this function?

1. ## 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

2. $\displaystyle f(x)= \frac{6}{x+2}-3$

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?

3. 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 $\displaystyle y$ , $\displaystyle y \ne -3$.

4. Hello, osmosis786!

When possible, I like to sketch a graph.

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

The graph of .$\displaystyle y \,=\,\frac{1}{x}$ .looks like this:
Code:
                  |
|*
|
| *
|  *
|    *
|           *
--------------+----------------
*           |
*    |
*  |
* |
|
*|
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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: .$\displaystyle y \,\neq\,-3$