A function is a relationship where for each and every value of the independent variable (here \(x\)) there is only one value of the dependent value (here \(y\)). If you can make a list of rules describing which value to match to each possible value of the independent variable value and know what to write down for the dependent variable, you have a function. If you have to include a rule like "if \(x\) is 3, then \(y\) is 4, except sometimes it is -2", you do not have a function.
Here for A = {(2, −2), (5, −5), (−2, 2), (−5, 5)} we can see that there is no question about the value of \(y\) that goes with any of the 4 values of \(x\) for which this relationship is specified. Therefore, A is a function.
For B = {(4, 2), (4, −2), (9, 3), (9, −3)} we are immediately able to see that we do not have a function, because we cannot make a simple list of rules to tell our indecisive helper how to respond when we give various values of \(x\). If we say \(x=4\), our indecisive helper would not know whether to respond with \(y = 2\) or \(y=-2\), and similarly, we cannot make a rule for them to follow for \(x=9\). Therefore, B is not a function.