Why is y=sqrt(x) not reflected in the x-axis
[From: ] [author: ] [Date: 12-04-22] [Hit: ]
the calculator assumes you only want the positive answers. If you want the negative values of y as well, you need a second graph of y = -sqrt(x). The reason for this is that if you include both the positive and negative sqrts, it is no longer a function. To be a function,......
EDIT:
For further note, if you're interested in mathematics, when you look at functions in most mathematics, unless otherwise specified they are only real valued functions. This means that for whatever input set you use you will always get real numbers in return. The graphs that you make are based off of real numbers for the x,y coordinates, so there is no way to 'map' the values outside of the real numbers, namely i (sqrt(-1)) and any number set larger than the reals (Complex numbers, quarternions)
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Because that wouldn't be a function. So mathematicians have chosen the square root notation sqrt(x) or x under a radical sign to stand for "the non-negative number whose square is x".
That's not the only function where it's convenient to choose one of more than one possible solution. There are infinitely many angles whose cosine is 1/2, but the arccos of 1/2 is only one number, the angle between 0 and 180 degrees whose cosine is 1/2. This is called the "principal value".
You're right of course, that there are two possible solutions to y^2 = x. If you're asking for solutions to that equation, you have to specify both. But if you're asking for "the square root of y", that's only one of the two solutions. We indicate the other one by writing -sqrt(y).
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y = sqrt(x) and y = -sqrt(x) are actually two separate functions. When you graph y=sqrt(x) in your calculator, the calculator assumes you only want the positive answers. If you want the negative values of y as well, you need a second graph of y = -sqrt(x).
The reason for this is that if you include both the positive and negative sqrts, it is no longer a function. To be a function, there can only be one value of y for each value of x. In other words, it doesn't pass the vertical line test. If you draw a graph of sqrt(x) and -sqrt(x) and draw a vertical line down the screen, you will cross the graph in two separate places, indicating it is not a function. For example, at x = 4, the y value equals both -2 and +2. A function cannot have two y values for any x, by definition.
So it is generally assumed you are only talking about the positive values of y when you write y=sqrt(x)
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Yes, what you say makes sense. However, sqrt(x) is usually (ie. by convention) taken to mean the positive square root. I think that's probably so that it is a function, which it wouldn't be if there were two values for the square root.
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y=sqrt(x) is NOT equivalent to y=±x. If you want it reflected over the x-axis, you have to graph y=-sqrt(x) which is a different function.
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It is not, their domain are different.
y = sqrt(x) is equivalent to x = y^2 when y > 0
y^2 = x
y = + sqrt(x)
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The sqrt of a negative number is given by sqrt(-x)= j sqrt(x).
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The square root cannot be negative...unless you're using imaginary numbers.
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