+701 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
+9479 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.)
Also, you might want to read the first paragraph of this page: https://en.wikipedia.org/wiki/Interval_(mathematics)
Hope this helps somewhat! If you're confused about something, please say so! 
+701 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
+118677 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 
+129852 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 !!!!
+701 i'm confused though,,,,
how does
\((-\infty,4)\) include \([-4,4)\)? is there a method of knowing this? :( Im sorry im really confused ahhh
+129852 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)
+701 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?
+701 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.
