Kim G. answered • 01/15/21
Yale student pursuing MA in statistics and PhD in public health
Hi Tara - great question! Set notation can be tricky and confusing.
1) Translate the set notation into normal English. To do this, I'd just basically take every symbol (or pair of symbols, i.e. parentheses) and make a sentence out of it. Here I'd say "the set of all points (x,y) where x is between sqrt(y) and 2, and y is between 0 and 4."
2) Based on your translation, determine how many dimensions your final drawing will be. Given that this statement starts out with "the set of all points (x,y)," I automatically know how many dimensions I'm working in - I'm working in the two-dimensional plane that I've been using since before multivariable calculus! If it said something like "the set of all points (x,y,z) such that . . . " or "the set of all points (x) such that, . . ." I know my final drawing would be three or one dimensions, respectively. But in this case, I know my drawing will be 2D.
3) Look for places where y depends on x, or vice versa. In this case, I see a y in the boundaries for x (since x cannot be less than sqrt(y)) , but I don't see any xs in the y boundary. I'll keep this in mind and start working with the easier boundary - the one without dependencies (in this case, the y).
4) Draw your unrestricted boundaries. The final part of the sentence we translated in Step 1 says that y must be between 0 and 4. That means that all points have a y coordinate between 0 and 4. A way to imagine that in pictures is to draw the lines y = 0 and y = 4. We know that the region we're looking for must be somewhere between those two. Also, draw the line x = 2: we'll figure out what to do with the other x bound in the next step.
5) Draw your restricted boundaries based on your unrestricted boundaries. This step may sound hairy, but hopefully I can help break it down. We want to know what values x can take that fall in the region we described in Step 1. The question I'm asking myself here is this: given that y is between 0 and 4, what are all possible values of sqrt(y)? That will tell us what our lower bound for x will be. (We already know the upper bound for x - it's 2).
The first thing I notice here is that the square root function is strictly increasing. (You can use your Calc 1 tools if you don't believe me). That means that over the entire domain [0,4], the function sqrt(y) is increasing. That's good! That means we can just use the end points to help us understand our domain for x. Note that sqrt(0) = 0, and sqrt(2) = 2. That means that the lower bound of x for this region looks like the square root function x = sqrt(y), meeting up with the line x = 2 at the very end. Go ahead and draw the function x = sqrt(y), and shade in the area that lies between all the lines you drew.
*NOTE: This last step asks you to draw the line x = sqrt(y), NOT y = sqrt(x). So take the image of the square root function that you're used to, and rotate it accordingly. Hint: If you're not exactly sure what this shape looks like rotated, try plotting a few points at, say, y = 0, y = 1/2, y = 1, y = 2, and y = 4.
Hope this helps! :)
