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Don’t you think that whoever defined functions as having to pass the vertical line test is kind of stupid?
The whole point of a function needing to pass that test is supposedly because it is said that a single input can not have two outputs. But that’s not true. If x=4, the square root of that could be 2 or -2. The only good reason I could think the square root function uses the absolute value symbol by convention is because people aren’t familiar with the fact that the square root of a number could be plus or minus.
2 Answers
- Random StrangerLv 62 weeks ago
You've got it backwards and the definition is not stupid, but carefully crafted.
It is not a "supposedly" condition. A function DOES only return one value. That's part of the definition of a function and is ALWAYS TRUE. Period. That fact leads to what is called the vertical line test.
The reason for using the absolute value with square roots is because people were well aware that there were two solutions to x^2 = <a positive value> but WANTED the square root to be a FUNCTION, returning only one value. So, the square root FUNCTION Is DEFINED to return the principle root, the positive value. Hence, √(x^2) = |x|, not ±|x|. Square root is a function and cannot return more than one value. Remember that as you go forward in math. Forgetting the square root is a function can lead to serious mistakes. Memorize that: SQUARE ROOT IS A FUNCTION and is not just the inverse of squaring.
If you are asked to find the SQUARE ROOT of a positive number then the answer will always be positive, by definition. If you are asked to find the numbers that when squared yield that positive number then there are two answers. Notice that the questions are fundamentally different.
You have to be aware of that while solving equations. If you have an expression that uses the square root FUNCTION such as √(x^2), then it can be replaced by |x|, a single value. If you are attempting to solve an equation that does not use the square root FUNCTION but involves squares, such as x^2 = A, then you have to take into account both of the possible values; in that case x = ±√A. You are temped to say that you are taking the square root of x^2 BUT YOU ARE NOT; you are actually just finding the two values ±√(x^2) or ±|x| which when squared yield A.
So, for example, x^2 = 4 has two solutions: +2 and -2. But, √((-2)^2) has only one solution. |-2| which is 2.
