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# Piecewise functions

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i struggle with piecewise functions as the way we are supposed to write the domain, range, int. of increase and decrease confuses me....

do we write it with curly braces or like interval notation,,,,and also what if you have an open circle and a closed circle on the x same x axis or two closed circles on the x axis (wouldn't that just not be a function)

thank you s o so soooosososososo much <3

if anyone can help me with these ill be eternally grateful

Oct 29, 2019
edited by Nirvana  Oct 29, 2019

#1
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We put one open curly brace beside the "pieces" of the piecewise function, for example:

$$f(x)=\begin{cases} x+2 & \text{if}&x> 2 \\ x^2 &\text{if}& -3< x\leq 2 \\ 9 &\text{if}& x\leq-3 \end{cases}$$

This is the way to show the pieces of a piecewise function.

On the other hand, domains and ranges (and, of course, intervals) can be expressed in interval notation.

An interval is a set of values or numbers - not a set of expressions or functions.

E.g., the interval  [0, 5)  is every number between  0  and  5, including 0 and excluding 5.

And by looking at the graph for this function: We can see:

 The domain is: _ (-∞, ∞) The range is: [0, ∞) The intervals of decrease are: (-3, 0) ← This means the function is decreasing for all x values in the interval  (-3, 0) For instance, when x = -1, the function is decreasing. The intervals of increase are: (0, 2)   and   (2, ∞)

All of those are intervals expressed in interval notation.

(At the points where  x = 2  and  x = -3 ,  the slope is undefined, and so the function is neither increasing nor decreasing, and so we do not include those values in the intervals of increase or decrease.)

If the graph passes the vertical line test, then it is a function.

Here are a bunch of examples: https://imgur.com/a/Px4lYIt

(If you still are wondering about whether a particular situation is considered a function or not, you could try to draw it on desmos and share the graph or draw it on a piece of paper and take a photo.)

Hope this helps somewhat! If you're confused about something, please say so! Oct 29, 2019
#5
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Oh my god,

you are an absolute life saver hectictar, and i hope you know it :D

Thanks to you, I can now do these without hesitations!!

but if it's okay may i ask a few more questions?

Alright,,,

So I have constructed the graph below and I don't understand the questions that i have written "on" the graph. If I could just get these questions answered, I will finally fully know this material :') Again, I just want to thank you so so so sooo so much for your help. I really appreciate it :D

Nirvana  Oct 29, 2019
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Thanks for putting such effort into explaining that Hectictar :)

I expect Nirvana, more then most, will take full advantage of your time and explanation.

You are both great assets to this forum Oct 29, 2019
edited by Melody  Oct 29, 2019
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WooooHOOOOOO Let's go Nirvana and Hectitar!!!

tommarvoloriddle  Oct 29, 2019
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Totally agree with Melody.....two  real assets    !!!   CPhill  Oct 29, 2019
#6
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Aw thank you Melody, Tommarvoloriddle, and CPhill!!
You guys are awesome!

and I'm not the one that really did anything,,, it was mostly Hectictar and her awesome explanation and clarifications which truly did help me and I appreciate her for it. :)

:D

Nirvana  Oct 29, 2019
edited by Nirvana  Oct 29, 2019
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Equation of line  with a constant increase

We have points (-4,3)  and  (4,8)

Slope  =  [8-3] / [ 4 - -4]  =  5/8

Equation is

y = (5/8)(x - 4) + 8

y = (5/8)x - 20/8 + 64/8

y = (5/8)x + 44/8

y = (5/8)x + 11/2

Equation of line with constant decrease

We have points  (4,4)  and  ( 9,1)

Slope  =  [ 1 - 4] / [ 9 - 4]  = -3/5

Equation is

y = (-3/5)(x - 4) + 4

y = (-3/5) + 12/5  + 4

y = (-3/5)x + 32/5

Based on your graph, we would have this  :

1                          if    -inf  < x  -4

f(x)  =       (5/8)x + 11/2         if    -4 ≤ x < 4

(-3/5)x + 32/5       if     4 < x  ≤ 9

1) and 2)

This is still "piecewise"......if we can draw a graph without lifting our pencil, it is continous (although it could still be  piecewise-continous).....your graph is "broken"...so...it is not piecewise continuous...

3)

This graph is continuous from    (- infinity, 4)

It is also continuous from  (4, 9 ]

Note that at x  = 4....the continuity is "broken"

Also note that the closed circle at x  = - 4   "picks up" the continuity from negative infinity

Hectictar can probably develop this even more   !!!!   Oct 30, 2019
edited by CPhill  Oct 30, 2019
#8
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thanks CPhill!

but how come during the inverval of continuity -4 isn't included?
like wouldn't [-4,4) be an interval of continuity? How come that's not included?

Nirvana  Oct 30, 2019
edited by Nirvana  Oct 30, 2019
#9
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[ -4, 4)   is included in the interval  (-infinity, 4)   CPhill  Oct 30, 2019
#10
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i'm confused though,,,,

how does

$$(-\infty,4)$$ include $$[-4,4)$$? is there a method of knowing this? :( Im sorry im really confused ahhh

Nirvana  Oct 30, 2019
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Note that  your graph  is  continuous  from negative infinity to  - 4.....but we have an open circle at  x  = -4

So.....if your graph stopped there  the interval of continuity would be (-infinity, -4)

However.....the "closed"  circle  at x  = -4  on the next function "fills in"  the continuity  which extends to "almost"  x  = 4

At x  = 4  the continuity stops

So.....your graph is actually continuous  from  (-infinity, 4)   CPhill  Oct 30, 2019
#12
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OHHHHH,
oh my god CPhill

THANK YOU SOOOO MUCH

you don't know how much you and hectictar just helped me...i appreciate it so much thank you thank you thank youuuuuuuu

Nirvana  Oct 30, 2019
#13
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OK....I hope I didn't confuse you too much.....LOL!!!!   CPhill  Oct 30, 2019
#14
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Nope!!
You did great, thank you so much :D

Nirvana  Oct 30, 2019
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I have a quick question if that's okay though,,,it has nothing to do with this....

does the U for union represent and or or?
if yes it is and, then what is the symbol for or? if no for and and it is or, then what is the symbol for and?

Oct 30, 2019
#16
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U   =   "or" = "union"

∩  =  "and" = "intersection"   CPhill  Oct 30, 2019
#17
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ohhhh okay thanks for clarifying

for some reason I was thinking that it meant and?

or is it only for special circumstances? or am i just going nuts LOL

and so....
when describing more thanTWO continuities, intervals of increase, decrease? do we use union as in or?
is that what they mean

let's say we have an int. of decrease thats (-5,6)u(8,9) something like that did i do that right??

and I'm SO sorry i keep asking questions it's probably getting annoying. im sorry.

Nirvana  Oct 30, 2019
#18
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No one is getting annoyed.  We like people to learn from us. That is a large part of why we want to be here.

People who do not ask qustions, do not learn.

Melody  Oct 30, 2019
edited by Melody  Oct 30, 2019
edited by Melody  Oct 30, 2019