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I am a high school junior and just enrolled in AP Calc AB. We are starting the review and we were going over range and domain. I was never taught domain and range and have always been behind. I have always used a cut-and-dry formulaic approach to math, I want a sure thing that I can apply to all types of problems. However, when approaching range, I find it hard to do this. I have heard things like graphing and taking inverse functions. I would love to graph but put simply some equations are too complex for me to the graph. Inverse functions as well. I understand that you can solve for range intuitively but I have a hard time doing so.

For example, find the domain and range of y=√x−4+5

The domain is simple for me, x≥4

Now we get to the range. This one is more simple. You can intuitively think that the √x−4 will either be equal to 0 or more. Therefore the smallest value of the range will be 5. As it can get infinitely bigger, the upper bound of the range will be positive infinity.

The problem is when I get to a more complex equation, I cannot intuitively understand what the range will be.

for example:

I struggle with x+2/x^2-4.

Domain, simple: x cannot equal plus or minus 2

Range: It is too complex for me to try to graph, I know you could simplify it to 1/x-2 but I still can't really graph that sadly. The inverse comes out to be (1/x)+2, meaning the range restrictions should be: x cannot equal 0. However, my math teacher told me using inverse functions will lead to complications and that that restriction was incorrect. So I am left with my seemingly last option to think about it intuitively. I think, "What are all the possible Y values for x+2/x^2-4. I am stuck here. I don't know where to even start to think about the possible restrictions of the range. Y can be undefined, Y can be 0, Y can be 1, etc.

How do I develop a better method of thinking about it? I realize this is a large problem with my mathematical thinking that I need to improve on. Any tips, books, or ideas are appreciated.

Theodore Veteto
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  • Didn't you just ask the same question? See https://math.stackexchange.com/questions/4519546/solving-for-range-intuitively#comment9493175_4519546 – John Douma Aug 27 '22 at 02:20
  • Yes someone suggested adding an example I didn't understand, I just don't know how to delete the other one. Haha – Theodore Veteto Aug 27 '22 at 02:21
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    Your formatting is off. Do you mean you struggle with $x+\frac{2}{x^2-4}$? – John Douma Aug 27 '22 at 03:02
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    "I still can't really graph that sadly" Graphing a complicated function is just about investigating it then piecing together various clues, and if there are still gaps, investigating further. By investigation, I mean to to ask questions like what the intercepts are, what happens as x gets very big, and then very small, what does it mean when an x-value makes the denominator 0, how does the curve's gradient vary, etc. Eventually, you will attain a better sense of how common functions look like, and any required investigation will speed up significantly. – ryang Aug 27 '22 at 03:33

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