Question

PLZ HELP!!
Given: 2f(x) + f(-x) = 3x,
Find: f(2)

Answer 1

did you try to replace the x's with 2's?

Answer 2

also do you know if f is even or odd?

Answer 3

f is a function

Answer 4

right do you know anything about its symmetry?

Answer 5

like is it even or odd?

Answer 6

nope

Answer 7

well try pluggin x=2
and also x=-2
giving you two equations

Answer 8

looks like you have a system of equations to solve

Answer 9

how does that give a system? i dont really know how to deal with functions sorry

Answer 10

plug in x=2: 2f(2)+f(-2)=6
plug in x=-2: 2f(-2)+f(2)=-6

Answer 11

We can solve for both f(2) and f(-2)
but we really only need f(2)
But we can solve for both for fun.

Answer 12

If f(2) and f(-2) look confusing to you... replace f(2) with a and f(-2) with b
--
2a+b=6
2b+a=-6

Answer 13

That should look a bit more familiar.
That is a system of linear equations

Answer 14

thanks, umm, b=2 a=2??????? no way

Answer 15

That doesn't give you that...

Answer 16

ok wait i redid it SORRY

Answer 17

b= -6

Answer 18

looks good

and b what f(-2)

Answer 19

so you found f(-2)
(and i'm trying to do this fighting a kitty while it bites me)

Answer 20

was f(-2)*
now we need our main objective which was finding a=f(2)

Answer 21

a=6

Answer 22

f(2) = 6?

Answer 23

looks adorable.

Answer 24

thanks. but how can i find the original f(x)?

Answer 25

becuase f(2) =6 does not imply that f(x) = 3x, right?

Answer 26

f(x) could be 3x
because 2(3x)+(-3x)=6x-3x=3x

Answer 27

is there a way to find out for certain what the function is, using the given information?

Answer 28

well f(x)=3x is one possible answer that satisfies 2f(x)+f(-x)=3x

Answer 29

right, but are there (can there be) others?

Answer 30

Maybe

Answer 31

what about \[f(x) = \left| x \right|\]

Answer 32

that works

Answer 33

f(2)=18 is other possiblity for even function here

Answer 34

how did you get that?

Answer 35

solve

\[2f(x)+f(-1)=3x\]
\[2f(-x)+f(x)=-3x\]

for \(f(x)\)

Answer 36

typo

2f(x)+f(-x)=3x

Answer 37

oh yeah
that looks gorgeous zarkon

Answer 38

opps i think i read it wrong since 3x is already given

Answer 39

how to solve that? very new w functions

Answer 40

multiply the first equation by -2 then add the equations

Answer 41

again rizags if it looks ugly like
that you could again replace f(x) with a and replace f(-x) with b

Answer 42

is it -f(-x) = 2f(x)?

Answer 43

nope, thats wrong

Answer 44

what exactly should i try to eliminate when solving?

Answer 45

2a+b=3x
a+2b=-3x
---
a+2b=-3x
-2(2a+b+3x)
------------
-3a+0=-9x
-3a=-9x
3a=9x
a=9x/3=3x
So I guess f(x)=3x

Answer 46

\[-2[2f(x)+f(-x)=3x]\Rightarrow-4f(x)-2f(-x)=-6x\]

now add
\[-4f(x)-2f(-x)=-6x\]

and

\[2f(-x)+f(x)=-3x\]

Answer 47

-3 f(x) = -9x, 3f(x) = 9x, f(x) = 3x

Answer 48

waiiit, then absolute value of x doesnt work?

Answer 49

|x| does not work

Answer 50

but it appears to because 2(2) + 2 = 3(2)

Answer 51

where x = 2

Answer 52

|x|=x if x>0
|x|=-x if x<0
if x>0 then we have 2x+-x=x not 3x
if x<0 then we have -2x+x=-x not 3x

Answer 53

let x=-2

Answer 54

ohhhhHhhhhhhhhhhh got it

Answer 55

if x>0 then it is true

\[2f(x)+f(-x)=2|x|+|-x|=2x+x=3x\]

Answer 56

wait, actually, if x = -2 then 2(2) + 2 is once again 3(2)

Answer 57

but you would want 3(-2)=-6

Answer 58

my mistake, i see now, thank you

Answer 59

Yes you're right @Zarkon

Answer 60

thanks sooo much

Answer 61

|-x|=x since x>0

Answer 62

yep

Answer 63

yep got it!

Answer 64

or i could have said |-x|=|-1*x|=|-1|*|x|=1*|x|=1|x|=x

Answer 65

where x>0

Answer 66

just wanted to by where x>0
so people know what I'm talking about :p

Answer 67

yep gotcha

Answer 68

thanks again!