How to Master the Techniques

To master the techniques that allow you to determine the domain, range, and graph, at least at this level, it is not necessary to review more theory than we have already covered. At this point, it is better to do exercises, and the best way to practice is by randomly inventing exercises. Only in this way will you truly know the limits of the techniques we have studied so far and develop the intuition that will allow you to perform with skill.

What we will do next is exactly that: I will invent some random exercises and solve them as the techniques reviewed permit; if they cannot be solved with them, I will explain at which points they fail and why.

Practice Exercises:

Determine the domain, range, and graph of the following functions:

1. a(x) = \displaystyle \frac{x^3 - 3x^2 + 5x - 1}{x^2 + 2x - 1}
2. b(x) = \displaystyle \frac{4x^4 + 2x^3 - 5x^2 - 2x - 2}{2x^2 - x - 1}
3. c(x) = \displaystyle \frac{x^5 + x^3 - x - 1}{x^2 - x - 1}
4. d(x) = \displaystyle \frac{3x^2 - 3x - 2}{\sqrt{x^2 - 1}}
5. e(x) = \displaystyle \sqrt{\frac{x^4 - x^2 - 11}{(x^2 - 1)\sqrt{x^2 - x - 1}}}
6. f(x) = \displaystyle \frac{(x^2 - 2x - 2)\sqrt{7x^8 - 5x^4 - 2}}{x\sqrt{5x^2 - 3x + 2}}

Practice Exercises Solution:

Proposed Exercises:

The following proposed exercises are similar to the ones solved earlier; I have only changed the numbers. The structure is the same, so the previous solutions may be useful. If you are not yet confident with these techniques, you can always rely on online tools like WolframAlpha and GeoGebra. If you are struggling with algebra, reviewing the following classes will help you:

As in the previous exercises, you must calculate the domain, range, and graph.

1. a(x) = \displaystyle \frac{-5x^3 + 9x^2 - 7x - 2}{5x^2 + 3x + 9}
2. b(x) = \displaystyle \frac{-8x^4 + 2x^3 + 4x^2 + 4x + 1}{3x^2 - 9x + 3}
3. c(x) = \displaystyle \frac{-7x^5 + 9x^3 + 7x + 5}{-2x^2 - 8x + 6}
4. d(x) = \displaystyle \frac{4x^2 - 4x - 9}{\sqrt{-3x^2 + 7}}
5. e(x) = \displaystyle \sqrt{\frac{9x^4 + 2x^2 + 7}{(-8x^2 + 4)\sqrt{-6x^2 + 9x + 5}}}
6. f(x) = \displaystyle \frac{(7x^2 + 6x - 1)\sqrt{9x^8 + 3x^4 + 9}}{5x\sqrt{3x^2 + 8x - 3}}

From here, if you want to practice more, the best thing to do is invent your own functions and try them out.