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Functions on the Real Line - Example 1

In mathematics, a function (or map) f from a set X to a set Y is a rule which assigns to each element x of X a unique element y of Y, the value of f at x, such that the following conditions are met: 1) For every x in X there is exactly one y in Y, the value of f at x; 2) If x and y are in X, then f(x) = y; 3) If x and y are in X, then f(x) = f(y) implies x = y; 4) For every x in X, there exists a y in Y such that f(x) = y.

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Okay, so let's get our hands dirty. Let's solve this equation now. Notice. There's an absolute value here. So we got to consider actually two cases. So the first case is that what's inside the absolute value is positive. So in other words, X minus two is equal to four. The other case is if we have a negative inside the absolute value, because that's going to be fine, too. And in that case, we'll have negative. X minus two is equal to four. Okay? And so we have this very important word. Or so either one of these equations could be true to make the initial equation true. So if we go through and solve it will get X is equal to six. So I'm just gonna add to over there and over here, let's multiply by negative one, so I'll have X minus two is equal to negative four. Or, in other words, X is equal to negative, too. So we actually have two solutions. X is equal to six is a solution, which you can verify or X is equal to negative two. So there are two solutions to this equation