How to determine the domain and range of this function? [closed]

I graduated several years ago from Computer Science. In order to refresh my mathematical knowledge, I bought the book "How to Think Like a Mathematician" and currently I'm working through the chapters. The author of the book provided the solution without the solution process. In most cases I don't know how to begin. Therefore, I need your hints for how to begin the solution process. So, for the following function I've to find out the domain. For training purposes I myself want to determine also the range.

$$f(x) = \frac{x}{x^2-5x+3}$$

For the domain of the function you have to check which values $x \in \mathbb{R}$ you can "put in" the function in order to get a result or better: such that the function is defined. The function is not defined if the denominator is zero. So you have to find the roots of the function $x^2 - 5x +3$.
• it turns out that I have to use quadratic equation. So the solution is $x \in R | x \neq \frac{-5 -\sqrt 13}{2} and x \neq \frac{-5 +\sqrt 13}{2}$. This was the first and last time I wrote with the iPad. It's exhausting to write with it this kind of text. – convexfx Sep 13 '16 at 21:21
• Your solution of the denominator function is correct. A nice way to write the domain could be $D = \mathbb{R} \setminus\{ \frac{-5-\sqrt{13}}{2}, \frac{-5+\sqrt{13}}{2} \}$. What do you have for the range of your function? – JDoe Sep 14 '16 at 7:42
• Hi JDoe, meanwhile I realized that I could to solve the function also by completing the square. But I don't get the same result as for the quadratic equation. Here is my solution process: 1. Bring function into complete square form -> $(x-a)^2$ $(x- \frac{5}{2})^2-( \frac{5}[2})^2+3$ 2. Combine constants $= (x-\frac{5}{2})^2-3,25$ 3. Take root $=$(x-\frac{5}{2}) - \sqrt 3,25 4. Rearrange and solve $x = 2,5 - \sqrt 3,25 or x = 2,5 + \sqrt 3,25$ So, I don't get the same result via quadratic equation. What am I doing wrong? – convexfx Sep 16 '16 at 22:15