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Thread: Math question (Reps)

post 1619526391 10-16-2020, 09:25 AM
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Math question (Reps)

When you graph the following function you get this:


But when you square both sides and then move y^2 to the left, you get this:


Why does the first equation get a half circle and the 2nd equation get a full circle when both equations are the same (just in a different format). Sorry if this seems basic but sometimes it's the basics that trip me up
post 1619526791 10-16-2020, 09:29 AM
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The second one isn't a function
https://en.wikipedia.org/wiki/American_decline
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post 1619526881 10-16-2020, 09:31 AM
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Originally Posted By BulkingIsHard⏩
The second one isn't a function
Could you elaborate?
post 1619527071 10-16-2020, 09:33 AM
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Originally Posted By anonkunbrah⏩
Could you elaborate?
Wikipedia:
In mathematics, a function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

https://en.wikipedia.org/wiki/American_decline
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post 1619527211 10-16-2020, 09:35 AM
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Originally Posted By anonkunbrah⏩
When you graph the following function you get this:


But when you square both sides and then move y^2 to the left, you get this:


Why does the first equation get a half circle and the 2nd equation get a full circle when both equations are the same (just in a different format). Sorry if this seems basic but sometimes it's the basics that trip me up
Think about it like this:

Plot the points that are possible given the formula.

In the first formula, what values of X are possible?

For example, when X = 2, that means Y = 0.
post 1619527391 10-16-2020, 09:37 AM
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Originally Posted By BulkingIsHard⏩
Wikipedia:
Didn't really help me get it but repped anyways
Originally Posted By SniXSniPe⏩
Think about it like this:

Plot the points that are possible given the formula.

In the first formula, what values of X are possible?
Actually i did that and i can see why it made the graph of a half circle. I think what threw me off is that doing something to both sides and rearranging an equation is supposed to result in the same thing. Yet doing that gets a different graph
post 1619527791 10-16-2020, 09:40 AM
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In the first:

If you discount imaginary numbers (j), the values of X can only be positive and the value of y cannot exceed 2 or be below -2 or else you end up with the squareroot of a negative number (and hence an imaginary number which can't be plotted here)

In the second:

both x and y can be positive or negative as you are squaring both but again neither can exceed +-2
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post 1619527871 10-16-2020, 09:41 AM
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Edit: Wizard ^^^

I think because in the first one x can never be a negative number as a result of a square root, unless it's imaginary: -1^1/2
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post 1619527901 10-16-2020, 09:41 AM
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post 1619528001 10-16-2020, 09:42 AM
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Originally Posted By AlexSays⏩
In the first:

If you discount imaginary numbers (j), the values of X can only be positive and the value of y cannot exceed 2 or be below -2 or else you end up with the squareroot of a negative number (and hence an imaginary number which can't be plotted here)

In the second:

both x and y can be positive or negative as you are squaring both but again neither can exceed +-2
this pretty much
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post 1619528231 10-16-2020, 09:44 AM
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Originally Posted By AlexSays⏩
In the first:

If you discount imaginary numbers (j), the values of X can only be positive and the value of y cannot exceed 2 or be below -2 or else you end up with the squareroot of a negative number (and hence an imaginary number which can't be plotted here)

In the second:

both x and y can be positive or negative as you are squaring both but again neither can exceed +-2
so it's because the square root is out of the picture that the restriction lets up and thus will allow for negative values of x? i was baffled since for everything else, it gets the same graph between function rearrangement (y=x-2 is same as x=y+2)
post 1619531341 10-16-2020, 10:22 AM
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The symbol for square root √ in mathematics is by convention the “principal root”....the nonnegative root.

Thus √(4) as written = 2. Not -2, or “plus or minus 2”......but 2. By convention.

So to answer your question.

The point (-2,0) does not satisfy your first equation x = √(4-y^2). It would be -2 = √(4-0^2)....but -2 cannot be the principal root of 4 (it’s negative).

The point (-2,0), however does satisfy the second equation x^2+y^2 = 4, and so is included in the graph.

If we actually solve for x in x^2+y^2 = 4 we get both halves x= √(4-y^2) or x= -√(4-y^2).

No need to bring up imaginary numbers etc, like some are doing above.
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post 1619537081 10-16-2020, 11:29 AM
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Originally Posted By anonkunbrah⏩
so it's because the square root is out of the picture that the restriction lets up and thus will allow for negative values of x? i was baffled since for everything else, it gets the same graph between function rearrangement (y=x-2 is same as x=y+2)
Squaring both sides can change the graph of an equation (better to use equation here rather than the word function).

For example, y=x is a basic line with slope 1, going through the origin.

However, y^2= x^2 is a double line, one with slope 1, the other with slope -1, intersecting at the origin. Making an “x”.

You are getting more possibilities with y^2 = x^2. Such as (-3,3), which fits this. The graph of an equation is all the points which satisfy the equation.

Notice that there are ramifications for squaring both sides of an equation: sure (-6)^2 = (6)^2, but this doesn’t mean -6=6.
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post 1619538111 10-16-2020, 11:41 AM
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To generalize the above, if you apply any many-to-one operation to both sides of an equation, then the resulting equation can possibly have solutions that are not solutions to the original equation.

For example

sin(x)=sin(0)
has many more solutions than just x=0.
post 1619538691 10-16-2020, 11:46 AM
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Originally Posted By AlexSays⏩
In the first:

If you discount imaginary numbers (j), the values of X can only be positive and the value of y cannot exceed 2 or be below -2 or else you end up with the squareroot of a negative number (and hence an imaginary number which can't be plotted here)

In the second:

both x and y can be positive or negative as you are squaring both but again neither can exceed +-2
could not have explained it better
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post 1619549521 10-16-2020, 01:48 PM
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Originally Posted By snailsrus⏩
could not have explained it better
Actually this is not really the explanation to OPs question. As mentioned above, x is nonnegative in the above since the first equation is the principal root. Not because of imaginary numbers.

Also see the above post about applying certain operations to both sides of an equation.
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