Square Root Function: Why Restrict Its Range to Non-Negative Numbers?
Date: 01/26/2010 at 01:58:08 From: name Subject: range of radical function negative or positive In a problem where you have to find the domain and range of a square root function, why is the range non-negative? For example, in f(x) = sqrt(x - 12), I know that the domain is all real numbers greater or equal to 12, but I am having trouble with the range part. My answer key says that the range is all non-negative real numbers, but it seems to me that it could be all non-positive real numbers instead. In my opinion, the range of a relation can be all real numbers. In y = sqrt(x - 12), where x is 48, y should work as both 6 and -6. After all, (6)^2 = 36, and (-6)^2 = 36, too. The graph is a curve that is symmetrical on the x-axis. A function has to pass the vertical line test, so one half of the graph has to go away. But why not preserve the negative half rather than the positive half? If you only have negative y-values for the function, the graph still passes the vertical line test. I realize that the solutions for a relation and a function would be different, because the function can only have 1 y-value assigned to an x-value. This means that the relation y = sqrt(x - 12) can't be a function because its graph has two halves, one above the x axis and one below, which fails the vertical line test.
Date: 01/26/2010 at 11:27:18 From: Doctor Peterson Subject: Re: range of radical function negative or positive Hi, Name. Yes, you COULD define a negative-square-root function instead of the positive-square-root function that we use; but in this setting, the radical symbol is defined to represent the principal root, which is defined as the non-negative one. Now, why DO we prefer the principal root to be non-negative? Is it just an arbitrary choice? No. One reason is that it makes the rules for working with a root simpler. You may recognize this rule: sqrt(ab) = sqrt(a) * sqrt(b) for any non-negative numbers a, b With the sqrt function defined as usual, that rule holds true: sqrt(4*9) = sqrt(36) = 6 sqrt(4)*sqrt(9) = 2 * 3 = 6 But if instead we had a negative (non-positive) root, which I'll call nsqrt to avoid confusion, we'd find this: nsqrt(4*9) = nsqrt(36) = -6 nsqrt(4)*nsqrt(9) = -2 * -3 = 6 So the general rule would have to be nsqrt(ab) = -nsqrt(a) * nsqrt(b) That's not as neat, is it? In math, we generally choose the definitions that make things as simple -- and as general -- as possible. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 01/26/2010 at 18:27:12 From: name Subject: Thank you (range of radical function negative or positive) Thanks for the fast and helpful answer.
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