Question

Hey guys, i need to prove that F(X+Y) = F(X) * F(Y).

Any help will be appreciated I will attach the actual question with this post.

Answer 1

question A

Answer 2

x= 1 or -1
y = -1 or 1

Answer 3

well it has to be a positive real number i think

Answer 4

the first property tells you that \(f(x+a)=cf(x)\) since \(f(x+a)\) has the same graph as \(f(x)\) but shifted right or left \(a\) units (depending on the sign of \(a\) )

Answer 5

use the second property to show that \(c=f(a)\)

Answer 6

i don't think that actually shows \(c=f(x)\)

Answer 7

Well since we know that f(0)=1 Let x=0 and f(x)=1
f(0+y)=f(y)=1*f(y)=f(0)*f(y)

Answer 8

ohhhhhh I think i understand what you're saying

Answer 9

it is always true that
\[f(y)=f(0+y)=1\times f(y)\]

Answer 10

and ya swissgirl that makes sense

Answer 11

Hey when you say: f(x+a)=cf(x) does c = a?

Answer 12

no sorry c = f(1) right?

Answer 13

Ok this isnt a proof but lets use a concrete example

Answer 14

Okay, any form of help is extremely appreciated. Thanks!

Answer 15

Also, I know that f(x) and f(x+a) will have similar graphs but is there a reason why? Honestly proving these kinds of things is a huge headache on me T__T !

Answer 16

Ok so let \(f(x)=e^x\) and so\( f(x+y)=e^{x+y}=e^y*e^x\)
So besides taht we have just shown that f(x+y)=f(x)*f(y) as you can see our c=e^y
Do you follow?

Answer 17

ya , cool i think i understand it

Answer 18

I had to prove this a while back but dont remember how to do it.
You gotta ask @satellite73 to do that for you :)

Answer 19

thanks for your help swissgirl : ) !!! This paper is due tomorrow and I'm not close to finish, so I really appreciate the help.

Answer 20

Wait r u in hs?

Answer 21

yah

Answer 22

sorry i don't want you to get the wrong idea that I intend to just cheat. I know a lot of this stuff, I just have difficulty proving it with the complicated questions.

Answer 23

nooo its fine. I can put it into words for you. I didnt realize it was hs level. Thought it was harder

Answer 24

Ok so before tell you what to write I just want you to get an understanding of what satellite was saying. Do you have a graphing calc?

Answer 25

yes i actually just did exactly what you did (the example you gave me) on the calc

Answer 26

i can't really explain why the graph is "scaled" as what the question asks for, i know that it gets pulled toward the 2nd quadrant but can't explain why

Answer 27

Ok so not sure if you know the exponents rule but \(e^x*e^3=e^{x+3}\)
So when multiplying exponents of the same base you keep the base and just add the exponents

Answer 28

yah

Answer 29

Now graph the function e^x and e^{x+3} on ur calc

Answer 30

k

Answer 31

if you will see both have the same shape BUT just shifted over. In mathematical terms we say one graph is a scalar of the other graph

Answer 32

i undesrtand

Answer 33

so basically it means its the same graph but it was multiplied by a constant
So for example c*f(x) where c is a constant is a scalar of the function f(x)

Answer 34

oh i think i understand

Answer 35

he actually made a note on this during class but i didn't bother copying it T___T

Answer 36

So basically your question said that f(x+y) is basically a scaled version of f(x) meaning that f(x+y)=c*f(x)

Answer 37

remember a scaled version of f(x) is c*f(x) where c is a constant

Answer 38

c is a constant? not f(y)?

Answer 39

Well we are going to show that c=f(y)

Answer 40

okay

Answer 41

@satellite73 whats the second property?

Answer 42

i don't think he's hear anymore ; (

Answer 43

Ya never know he may reappear :)

Answer 44

okay well thanks so much swissgirl

Answer 45

i'm going to give you a best response, i'm not sure if that means anything to you but i want to do it

Answer 46

i have a whole bunch of questions to answer still, hopefully before 2 a.m., i have to wake up at 6 haha

Answer 47

Well thsi question isnt fully answered but u will get partial credit for saying f(x+y)=cf(x) also If you show that f(0+y)=f(y)=1*f(y)=f(0)+f(y)
Then you can say let f(x)=e^x then f(x+y)=e^{x+y}=e^y*e^x =c*f(x) where c=e^y

Answer 48

okay thanks i think your example with exponents was very good

Answer 49

I have got load of hmwrk to do tooo. Doubt i will get to bed b4 2 am. Its like I study my whole life. Good Luck :)