Calculus 12 Question! Please wait for the attachment.

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Calculus 12 Question! Please wait for the attachment.

Mathematics
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Please explain the whole solution. I don't understand it.
where exactly are you stuck on the solution?

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Other answers:

I understand the slope and that the line passes those points. However, how to get the formula and the piecewise function. I don't understand. Could you please clarify it for me?
they broke the function into 3 linear pieces each piece is really a full linear equation, but restricted on the domain
|dw:1443244678300:dw|
|dw:1443244692570:dw|
ok
|dw:1443244719667:dw|
|dw:1443244752626:dw|
one question
go ahead
for the second lienear function f(x)= -x +2 why 2 is not the y-intercept?
extend out the middle piece in both directions you'll see that piece cross 2 on the y axis
|dw:1443244906871:dw|
right :D
one more thing, for the third function. why don't we say y=0 when x greater or equal to 2. Because at 2, y is 0
notice the \(\Large 1 < x \le 2\) as the restriction on the second piece if x = 2, then you use y = 2-x to find y if we used \(\Large x \ge 2\) for the third piece, then that would mean x = 2 is defined for the third piece as well. Question is: which piece do we pick if x = 2? This ambiguity is a bad thing when it comes to functions. So that's why the restrictions are set up in a way to avoid letting x = 2 be defined for more than one piece
put another way, the fancy notation you see at the bottom really breaks down into this f(x) = x if 0 <= x <= 1 OR f(x) = 2-x if 1 < x <= 2 OR f(x) = 0 if x > 2 each piece can't overlap another piece. Otherwise, we wouldn't have a function
what do you mean by overlap?
I believe connected does not mean overlapped
by overlap, I mean defined in multiple places
Oh I see, and the last question. Can we say we have 3 formulas for the function?
Is the formal way to put the solid point on the left function, or can you do whatever for the intervals
Ok thank you @DanJS
think of the 3 restrictions as number lines|dw:1443245579767:dw|
oh i didnt see tha tin the book , i was just wondering
now if we had x >= 2 for the third piece, then we have this overlap |dw:1443245694279:dw|
Right, hollow ones mean the number itself does not include
So if both are full. means overlapped becaue the numbers are present in two different equations.
exactly
@jim_thompson5910 thank you my friend. I felt like I was in a classroom. lol
is it legal to reverse each solid - dot combo in between functions?
** solid - open
i am guessing yes, since it describes the same thing, maybe
`is it legal to reverse each solid - dot combo in between functions?` I'm not sure what you mean. You mean use "solid" to mean "open/excluded endpoint" ? If so, then you're going against all of the conventional notation I've seen. That would be like making "up" be "down"
I think that step function wasopen - solid
typo corrected in next line, i just mean can you specify the intervals another way, for each interface between functions, can you choose arbitrarily which function will include the value
oh, yeah you could say maybe 0 <= x < 1 1 <= x < 2 x >= 2 just as long as there is no overlap, you're fine
k, that is what i was trying to say, solid-open endpoint combo switch deal.. blahh
of course if that value is defined for both functions

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