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
$\displaystyle f(x)= \frac{6}{x+2}-3$
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?
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 $\displaystyle y$ , $\displaystyle y \ne -3$.
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:Domain: .$\displaystyle x\,\neq\,0$Code:| |* | | * | * | * | * --------------+---------------- * | * | * | * | | *| |
. 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: .$\displaystyle y \,\neq\,-3$