Question

How do I solve piecewise functions? I cannot tell you how many times I have stared at my notes and done the practice.

https://gyazo.com/8740807019fb8b146a7ec27a3565554e

Answer 1

f(-2) can be evaluated by the piece that includes x=-2

Answer 2

is -2 in the set of numbers that is described
by x>=0?
by x<-3?
or by -3<=x<0?

Answer 3

Yes? the last one?

Answer 4

yes the last one -2 is between -3 and 0

Answer 5

so you use f(x)=2 since x is between -3 and 0

Answer 6

which means f(anything in that interval)=2

Answer 7

f(-1)=2
f(-1/2)=2
f(-2.534)=2

Answer 8

so that means my answer is 2?
So okay, the way I see is if the f(-2) or whatever number it is, happens to be in the set of functions, then the answer to f(x) is the opposite?

Answer 9

what?

Answer 10

I used f(x)=2 since x is between -3 and 0

Answer 11

if it said evaluate f(-5)
I would have used f(x)=|x+3| since x is less than -3

Answer 12

f(-5)=|-5+3|=|-2|=2

Answer 13

so it has to fit in the thing? i'm confused now.

Answer 14

remember the first thing we looked at is what set of x values our x was included in
then we used the corresponding piece

Answer 15

another example if we wanted to evaluated f(-10)
-10 is less than -3
and our function says use f(x)=|x+3| if x is less than -3
so f(-10)=|-10+3|=-7|=7

Answer 16

another example if we wanted to evaluate f(20)
well 20 is greater than 0
and our function says to use f(x)=x if x greater than or equal to 0
so f(20)=20

Answer 17

You lost me.

Answer 18

oh which part?

Answer 19

f(-10) or f(20) or what?

Answer 20

which means f(anything in that interval)=2

Answer 21

Like I said, I really don't get these piecewise functions..

Answer 22

before I said that
I said this "so you use f(x)=2 since x is between -3 and 0"
the interval being [-3,0) actually since it says to include -3

f(anything in that interval)=2

means any x value in [-3,0) will give us the output 2 when put into our function
examples
f(-3)=2
f(-2.9999)=2
f(-2.5)=2
f(-2)=2
f(-1.99)=2
f(-1.5)=2
f(-1/2)=2
f(-1/5)=2

Answer 23

but how did you get 2 from -2?

Answer 24

Your function says to use f(x)=2 if -3<=x<0

Answer 25

and -2 certainly falls in that interval

Answer 26

f(-2)=2 since -3<=-2<0

Answer 27

Okay, so that little comma thing is like "if this answer is right, use "2" for x "

Answer 28

no x is -2 in your problem
f(x) or y is 2 for your problem

Answer 29

if it did say evaluate f(2)
then you would look at the inequalities (or intervals whatever you want to call them)
and see which one it satisfies
2>=0 so we use f(x)=x since x>=0 for this example
so f(2)=2

Answer 30

does this still make no sense?

Answer 31

your function says:
if x>=0 then use f(x)=x
if x<-3 then use f(x)=|x+3|
if -3<=x<0 then use f(x)=2

Answer 32

find which if part is true for your x then use function that corresponds to that inequality

Answer 33

to find the output for the function

Answer 34

x = -2 and y = 2

Answer 35

then what do I do with all these commas?

Answer 36

You don't need to do anything with them

Answer 37

if you want you can replace them with the word if
if you prefer that

Answer 38

but you already found f(-2)

Answer 39

okay, then what do I do with it?

Answer 40

what does it refer to?

Answer 41

what third bit?

Answer 42

I'm absolutely lost on what we are talking about now

Answer 43

is there another question or something?

Answer 44

The only question I see is find f(-2)
and we done that

I don't see a third or even a second question on the picture you posted

Answer 45

but what did we find? I'm confused on what we actually found.

Answer 46

we found f(-2)

Answer 47

and what is f(-2)? 2?

Answer 48

yes
f(-2)=2

Answer 49

remember if we have -3<=x<0 then we use f(x)=2
and we had x=-2 was included in -3<=x<0 so we used f(x)=2 to find f(-2)

Answer 50

insert any number from the inequality -3<=x<0 into f(x)
and you get the output 2

insert any number from the inequality x>=0 into f(x) you get the output x

insert any number from the inequality x<-3 into f(x) you get the output |x+3|

Answer 51

so my answer to this problem is 2, which I think is C?

Answer 52

So I guess you are asking me this question for the 4th time because you still don't understand why f(-2)=2?

Answer 53

pretty much.

Answer 54

Ok do you understand the below?

\[f(x)=x \text{ if } x \ge 0 \\ f(x)=|x+3| \text{ if } x<-3 \\ f(x)=2 \text{ if } -3 \le x <0 \]

Answer 55

we plugged in -2

we are looking for f(-2)

we replaced x with -2

replace all the x's with -2


\[f(-2)=-2 \text{ if } -2 \ge 0 \\ f(-2)=|-2+3| \text{ if } -2<-3 \\ f(-2)=2 \text{ if } -3 \le x <0\]

the only line that is true here is the last line
do you see why?

Answer 56

*\[f(-2)=-2 \text{ if } -2 \ge 0 \\ f(-2)=|-2+3| \text{ if } -2<-3 \\ f(-2)=2 \text{ if } -3 \le -2 <0\]*

Answer 57

the only line that is true here is the last line
do you see why?

Answer 58

the last line is the only line that is true since:
\[f(-2)=-2 \text{ if } -2 \cancel{\ge} 0 \\ f(-2)=|-2+3| \text{ if } -2\cancel{<}-3 \\ f(-2)=2 \text{ if } -3 \le -2 <0\]

Answer 59

yes, i get why we have 2. kinda. but i'm not sure what to do with it. Is it the answer to the problem?

Answer 60

the last line says f(-2)=2 ...

Answer 61

which makes 2 our answer. so the answer to the original problem is 2.

Answer 62

did you want to try more examples?
if so see if you can find f(-100)