why is this wrong?

allegansveritatem

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I have numbered the steps in this transformation so that it will be easier to follow(I hit upon this technique today while working on an equation and found it so useful in the way intended that I will follow this procedure henceforward.)I know this below is wrong--the signs should be reversed--but I can't for the life of me see why it is wrong. I went through it at least 4 times and couldn't find the error. Can anyone point it out?sinx0501.PNG
 
Show all the steps you did from equation 5 to equation 6.
 
If a+b=0 then -a-b =0. Yes?
Multiply both sides of your final equation by -1.
That doesn't mean your answer is incorrect, just in a different form to the answer you have been given.
 
Show all the steps you did from equation 5 to equation 6.
after 5 what I did was mulltiply the three terms on the left by 2 in order to get rid of the fraction. On the right the multiplier was swallowed up by the zero.
 
If a+b=0 then -a-b =0. Yes?
Multiply both sides of your final equation by -1.
That doesn't mean your answer is incorrect, just in a different form to the answer you have been given.
yes, but when I enter the function in my graphing calculator after having multipied it by minus one, I get a different picture than when I don't. Therefore, something has happened, no?
 
yes, but when I enter the function in my graphing calculator after having multipied it by minus one, I get a different picture than when I don't. Therefore, something has happened, no?
Let's say you have the equitation 2x=0. Multiple it by 2: 4x=0
Graph 2x and 4x - of course they are different. This is expected. After the multiplication the equation remained valid, but the expression on the left changed.
 
yes, but when I enter the function in my graphing calculator after having multipied it by minus one, I get a different picture than when I don't. Therefore, something has happened, no?
From this thread and your preceding thread (which by the way you never completed responding to), I now see a part of your normal thought process that is sometimes leading you astray. You are extrapolating from your graphing calculator in inapprpriate ways. Graphing calculators are wonderful tools, but they must be used properly.

[MATH]h(x) = 2 * f(x)[/MATH] means that, in every case but one trivial one, h(x) is a different function from f(x), and your graphing calculator will show that.

Except in two trivial cases, [MATH]h(x) = g(x) * f(x)[/MATH] means that h(x) is a different function from f(x), and your graphing calculator will show that.

BUT that does not mean that if u is one root of f(x) and h(x) = g(x) * f(x) that u is not a root of h(x). Here is a theorem that has no exceptions whatsoever

[MATH]h(x) = g(x) * f(x) \implies \text {the set of roots of } h(x) = \text {the set of roots of } g(x) \text { and } f(x) \text { combined.}[/MATH]
Don't let the fact that the graphs of two functions look different overall obscure the fact that they may be identical in one or more key respects. Your graphing calculator shows you everything, but it will not tell you what is relevant.
 
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If you are thinking about an equation of the form f(x) = 0 by graphing the function f, then all you need to look at on the graph is the x-intercepts -- that is, where the function equals zero. If you compare this equation to another, g(x) = 0, then as long as both graphs cross the axis in the same places, the equations are equivalent, though the functions are different.
 
The equations have identical solutions would be how I say what Dr. Peterson says in the previous thread, but both statements mean the exact same thing.
 
From this thread and your preceding thread (which by the way you never completed responding to), I now see a part of your normal thought process that is sometimes leading you astray. You are extrapolating from your graphing calculator in inapprpriate ways. Graphing calculators are wonderful tools, but they must be used properly.

[MATH]h(x) = 2 * f(x)[/MATH] means that, in every case but one trivial one, h(x) is a different function from f(x), and your graphing calculator will show that.

Except in two trivial cases, [MATH]h(x) = g(x) * f(x)[/MATH] means that h(x) is a different function from f(x), and your graphing calculator will show that.

BUT that does not mean that if u is one root of f(x) and h(x) = g(x) * f(x) that u is not a root of h(x). Here is a theorem that has no exceptions whatsoever

[MATH]h(x) = g(x) * f(x) \implies \text {the set of roots of } h(x) = \text {the set of roots of } g(x) \text { and } f(x) \text { combined.}[/MATH]
Don't let the fact that the graphs of two functions look different overall obscure the fact that they may be identical in one or more key respects. Your graphing calculator shows you everything, but it will not tell you what is relevant.
I will have to think this out. But I see a little bit of light up ahead. As for the previous post, I was not aware I had responded fully. I will go back and check it tonight. Thanks for this explanation. I will have to relate it to the case at hand.
 
If you are thinking about an equation of the form f(x) = 0 by graphing the function f, then all you need to look at on the graph is the x-intercepts -- that is, where the function equals zero. If you compare this equation to another, g(x) = 0, then as long as both graphs cross the axis in the same places, the equations are equivalent, though the functions are different.
This is what I am beginning to see. I looked at the curves and not at the x intercepts.
 
The equations have identical solutions would be how I say what Dr. Peterson says in the previous thread, but both statements mean the exact same thing.
OK. I was taken in by appearances and didn't go to the heart of the matter. To paraphrase Bill Clinton, "It's the intercepts, stupid!" It seems to me I know this at one point, and then it slipped into the "great Id" and got lost there.
 
Let's say you have the equitation 2x=0. Multiple it by 2: 4x=0
Graph 2x and 4x - of course they are different. This is expected. After the multiplication the equation remained valid, but the expression on the left changed.
I haven't got my calculator here but I will check this out. I need time to think a bit on this stuff. "The light shineth in darkness and the darkness seeth it not", but I think I see some rays beginning to break through. I will get back to the thread.
 
So a bit of intellectual autobiography.

When I went to college, way way back when mastodons still roamed in what is now Central Park. the university that I went to then required that every student take (a) at least four semesters of a physical science, or at least four semesters of mathematics, or at least two semesters of a physical science and at least two semesters of mathematics, (b) at least two semesters on the western canon of literature, (c) at least one semester on western music, (d) at least one semester on western art, and (e) at least four semesters on western philosophy, economics, or political theory. If you graduated from that college, you knew enough pure theory to be wildly dangerous in every major area of human endeavor, but at least you had some inkling about western culture before you ventured out into parts unknown.

When I first matriculated, I was planning on becoming a chemical engineer (what WAS I thinking) while most definitely taking a minor in beer and girls. So I took qualitative chemistry and real analysis for the a requirement. I have never had so miserable a time intellectually as I did in real analysis. I kept fighting what I found the ugly and anti-intuitive basis of standard analysis and trying instead to develop my own brand of analysis. Ironically, at the exact same time, Robinson was developing non-standard analysis at Yale. I just went to the wrong school for analysis, but I was only 17 and knew nothing about anything. Then, as a sophomore, I took organic chemistry. OMG: memorization on memorization. So I completely switched to European history and languages (while retaining the minor in beer and girls). Understanding barons and Machiavelli proved invaluable in a business career.

Now I have no bent whatsoever toward religion, but I do love poetry and myth. And in the course on the western canon of literature, we started with Genesis, Job, Mathew, and John. Job and John are great poetry, and Genesis is myth on the grand scale. John starts with an allusion that, in the original Greek, is also a pun. Early in Genesis, the Hebrew says "God says 'Let there be light.'" But John starts with a verse that includes the word "logos" three times and the word theos (inflected twice). "Theos" means a god, but "Logos" has a very wide field of meaning in Greek. One meaning is "word," alluding to Genesis. But another meaning is "logic, reason." Light = reason. And I could believe in a religion that deified reason. So your quotation from John appeals to me (not that I am trying at all to say that John is reasonable overall).
 
So a bit of intellectual autobiography.

When I went to college, way way back when mastodons still roamed in what is now Central Park. the university that I went to then required that every student take (a) at least four semesters of a physical science, or at least four semesters of mathematics, or at least two semesters of a physical science and at least two semesters of mathematics, (b) at least two semesters on the western canon of literature, (c) at least one semester on western music, (d) at least one semester on western art, and (e) at least four semesters on western philosophy, economics, or political theory. If you graduated from that college, you knew enough pure theory to be wildly dangerous in every major area of human endeavor, but at least you had some inkling about western culture before you ventured out into parts unknown.

When I first matriculated, I was planning on becoming a chemical engineer (what WAS I thinking) while most definitely taking a minor in beer and girls. So I took qualitative chemistry and real analysis for the a requirement. I have never had so miserable a time intellectually as I did in real analysis. I kept fighting what I found the ugly and anti-intuitive basis of standard analysis and trying instead to develop my own brand of analysis. Ironically, at the exact same time, Robinson was developing non-standard analysis at Yale. I just went to the wrong school for analysis, but I was only 17 and knew nothing about anything. Then, as a sophomore, I took organic chemistry. OMG: memorization on memorization. So I completely switched to European history and languages (while retaining the minor in beer and girls). Understanding barons and Machiavelli proved invaluable in a business career.

Now I have no bent whatsoever toward religion, but I do love poetry and myth. And in the course on the western canon of literature, we started with Genesis, Job, Mathew, and John. Job and John are great poetry, and Genesis is myth on the grand scale. John starts with an allusion that, in the original Greek, is also a pun. Early in Genesis, the Hebrew says "God says 'Let there be light.'" But John starts with a verse that includes the word "logos" three times and the word theos (inflected twice). "Theos" means a god, but "Logos" has a very wide field of meaning in Greek. One meaning is "word," alluding to Genesis. But another meaning is "logic, reason." Light = reason. And I could believe in a religion that deified reason. So your quotation from John appeals to me (not that I am trying at all to say that John is reasonable overall).
I could as well have quoted, who was it? Bach's? or Goethe's? last words: "Light, more light!" As for being bent toward or away from religion, I don't know which I am, but I do know that as I grow old I more and more appreciate the ten commandments and wish more and more that I had done something with them other than find good reasons for infringement and disregard. "Again bite of inwit" An old guy's remorse.
 
From this thread and your preceding thread (which by the way you never completed responding to), I now see a part of your normal thought process that is sometimes leading you astray. You are extrapolating from your graphing calculator in inapprpriate ways. Graphing calculators are wonderful tools, but they must be used properly.

[MATH]h(x) = 2 * f(x)[/MATH] means that, in every case but one trivial one, h(x) is a different function from f(x), and your graphing calculator will show that.

Except in two trivial cases, [MATH]h(x) = g(x) * f(x)[/MATH] means that h(x) is a different function from f(x), and your graphing calculator will show that.

BUT that does not mean that if u is one root of f(x) and h(x) = g(x) * f(x) that u is not a root of h(x). Here is a theorem that has no exceptions whatsoever

[MATH]h(x) = g(x) * f(x) \implies \text {the set of roots of } h(x) = \text {the set of roots of } g(x) \text { and } f(x) \text { combined.}[/MATH]
Don't let the fact that the graphs of two functions look different overall obscure the fact that they may be identical in one or more key respects. Your graphing calculator shows you everything, but it will not tell you what is relevant.
Yes, sometimes functions are only identical at the x axis and sometimes only at the y axis....and sometimes they intercept elsewhere and so it goes. By the way, I am not finding the thread where I have not fully responded. What is the title?
 
I could as well have quoted, who was it? Bach's? or Goethe's? last words: "Light, more light!" As for being bent toward or away from religion, I don't know which I am, but I do know that as I grow old I more and more appreciate the ten commandments and wish more and more that I had done something with them other than find good reasons for infringement and disregard. "Again bite of inwit" An old guy's remorse.
Goethe, a creature very much of the Aufklarung
 
Goethe, a creature very much of the Aufklarung
yes, Goethe. I sometimes confuse the author of this quote with Bach, probably because poor Bach was almost blind when he died and it would be logical of him to ask for more light even in extremis.
 
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