What is the best way to find the range of a function? For example if we have a function given: $$y = \frac{\sqrt{x+3}}{x^3 - 2x^2 - 8x}$$

The domain is easy enough to find: Ensure that the term inside the square root is positive and that the denominator is not zero. However, how would we find the range of the function? Apart from graphing the function. I would like to do it algebraically.


In general that's not easy to do. Graphing is a very good idea.
Well, to do it algebraically:

  1. The numerator is defined for $x \ge -3$ (presuming you want to stick to real values).
  2. The denominator factors as $x (x +2) (x-4)$.
  3. For $-3 \le x < -2$, $y \le 0$, going continuously from $0$ at $x = -3$ to $-\infty$ as $x$ approaches $-2$ from the left. So the interval $(-\infty, 0]$ is included.
  4. For $4 < x < \infty$, $y > 0$, going continuously from $+\infty$ as $x$ approaches $4$ from the right to $0$ as $x$ approaches $+\infty$. So the interval $(0,+\infty)$ is included.
  5. Putting these together, the range is the whole real line.

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