Hi I'm doing an engineering course and i've got stuck on this question.
f(x)= x/x^2-5x-6
I've worked out the domain to be x is not = to -1, x is not equal to 6
but I'm struggling to work out the range
The correct way to write this function is x / (x^2 - 5x - 6). I recommend reviewing the order of operations.
It helps to sketch the graph of f(x) to come up with hypotheses to prove regarding the range.
Hello, atpod1!
As emakarov pointed out, a graph would illustrate the behavior of this function.
$\displaystyle f(x) \:=\: \frac{x}{x^2-5x-6}$
I've worked out the domain to be: $\displaystyle x \ne \text{-}1,\;x \ne =6.$
but I'm struggling to work out the range.
There are vertical asymptotes at $\displaystyle x = \text{-}1$ and $\displaystyle x = 6.$
Plot a few points and we find that:
. . The graph contains the origin $\displaystyle (0,0)$
. . To the far right and far left, the graph approaches the x-axis.
. . To the left of $\displaystyle x = \text{-}1$, the graph is below the x-axis.
. . On $\displaystyle (\text{-}1,0)$, the graph is above the x-axis.
. . On $\displaystyle (0,6)$, the graph is below the x-axis.
. . To the right of $\displaystyle x = 6$, the graph is above the x-axis.
The graph looks like this:
It appears that $\displaystyle f(x)$ will take on all values.Code:| : | : :* | :* : | : : | : * : * | : * : | : * : * | : * : * | : * - - - - - - - + - - * - - + - - - - - - - - - - * -1: | * :6 * : | * : * : | : * : | * : : | : * : | : : | *: : | : *: | : : | :
Range: .$\displaystyle (\text{-}\infty,\:\infty)$