# Thread: Finding the range of a function

1. ## Finding the range of a function

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

2. ## Re: Finding the range of a function

$\frac{x}{x^2}-5x-6=\frac{1}{x}-5x-6$. Its domain is ℝ \ {0} and its range is ℝ.

3. ## Re: Finding the range of a function

everything sits under x as in

x divide x^2-5x-6

4. ## Re: Finding the range of a function

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.

5. ## Re: Finding the range of a function

Hello, atpod1!

As emakarov pointed out, a graph would illustrate the behavior of this function.

$f(x) \:=\: \frac{x}{x^2-5x-6}$

I've worked out the domain to be: $x \ne \text{-}1,\;x \ne =6.$

but I'm struggling to work out the range.

There are vertical asymptotes at $x = \text{-}1$ and $x = 6.$

Plot a few points and we find that:

. . The graph contains the origin $(0,0)$

. . To the far right and far left, the graph approaches the x-axis.

. . To the left of $x = \text{-}1$, the graph is below the x-axis.

. . On $(\text{-}1,0)$, the graph is above the x-axis.

. . On $(0,6)$, the graph is below the x-axis.

. . To the right of $x = 6$, the graph is above the x-axis.

The graph looks like this:

Code:
                      |
:     |     :
:*    |     :*
:     |     :
:     |     : *
: *   |     :  *
:     |     :    *
:  *  |     :       *
:   * |     :             *
- - - - - - - + - - * - - + - - - - - - - - - -
*       -1:     | *   :6
*     :     |  *  :
*   :     |     :
*  :     |   * :
:     |     :
* :     |     :
:     |    *:
:     |     :
*:     |     :
:      |     :
It appears that $f(x)$ will take on all values.

Range: . $(\text{-}\infty,\:\infty)$

6. ## Re: Finding the range of a function

Thank you Soroban, your help is much appreciated