Why is there missing solutions here

Question

As we can see there is a missing portion in the purple graph, Why do we have missing solutions here?

Also happens in other graphing calculators. I feel like we should get all the solutions but why are some parts missing?

Answer 1

This has to do with numerical algorithms to evaluate the function and plot the curve. One simple algorithm (that you may have learned in school) basically involves selecting various $x$-values and then calculating the corresponding $y$-values, then connecting adjacent points with a curve. But a computer does this more rapidly than a human, so it can select many more points to evaluate.

But a computer doesn't handle points at which the evaluation of the function no longer yields a real number: if $|x| > 1$, then $\pm \sqrt{1 - x^2}$ is an imaginary number. So in the neighborhood of $x = 1$, the computer is happily plotting values like $x = 0.99351$, but then at the next step, it might choose $x = 1.00537$ and suddenly it encounters a result it cannot plot. So there is no way for it to joint the previous point to the next one. This is why the curve stops before $x = 1$, because the algorithm would somehow need to know that $x = 1$ is the largest value for which $y$ is defined. But in practice, it is not generally possible for a computer to know this, because it involves solving for the inverse function.

What more sophisticated algorithms do instead is to leverage the available computing power, and in regions with discontinuities or irregular behavior, it will adaptively subsample the neighborhood--choosing finer increments for the $x$-values so that it can gather more information about the behavior of the function at that point. But this comes at some computational cost, and even doing this may not adequately "close" the curve--if we zoom in further (as you show in your first picture), the computer also has to compensate by evaluating more points.

So, why does the red curve close when the purple and green ones do not? This has to do with the choice of algorithm. In the red curve example, you write $$y^2 = (1.1)^2 - x^2.$$ This way of writing the equation allows the use of a different algorithm to find the behavior of the relationship between $x$ and $y$. For example, a crude method is to select values for both $x$ and $y$ and see how closely they satisfy the equation. What I mean by this is, the computer might rewrite the equation as $$f(x,y) = x^2 + y^2 - (1.1)^2,$$ and try to detect where the function changes sign, since the curve lies where $f(x,y) = 0$. For instance, the computer might select $(x,y) = (0.9, 0.2)$. If the result is close to $0$, then it knows to search more in that area. If it is off, then the computer knows to search elsewhere. It does this for many ordered pairs in the viewing region, successively refining the guesses until the error is below a certain threshold. Because the guesses don't result in imaginary numbers, and the sign of the guess also gives some indication as to which direction to search, this algorithm does not encounter the kind of problems that the first one does. To be sure, it has other problems, but a detailed discussion is beyond the scope of this answer.