An explanation of how to find the domain, range, vertical and horizontal asymptotes just in general. Thanks.
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The domain of an equation is a valid input. So invalid inputs need to be eliminated. So roots and division can cause some issues there. Once you know the valid input, that can be used to find the output, or range.
If the bottom of a fraction approaches 0, there may be a vertical asymptote or a removable hole. Which depends on if the zero cancels out with something on the top. So the zeros that do not cancel out when you factor are all places where asymptotes can live.
Horizontal asymptotes have to deal with end behavior. When numbers get super positive and super negative, what happens? There are a few ways to test this. One, divide through by the highest exponent. That is best shown:
Now, as x goes to \(\pm\infty\) those fractions become zero, and the end behavior is that it goes towards 1/2.
Another way is to graph it. This is always a good double check because some end behaviors are deceptive, especially if a root is involved.
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horizontal asymptotes and range are directly related
vertical asymptotes and domain are directly related
any position the domain is excluded (for instance, x != 2), you will have a vertical asymptote, meaning you can approach 2 very closely, but never equal it
any position the range is excluded (for instance, f(x) != 0), you will have a horizontal asymptote, meaning you can approach the x-axis very closely, but never equal it.
not really "definitions", but hope it helped