SOLUTION: If f(2x+1)=6x+2 what is f(x)

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Question 1149579: If f(2x+1)=6x+2 what is f(x)
Found 3 solutions by Edwin McCravy, jim_thompson5910, ikleyn:
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
f(x) = 6x+2

To find f(2x+1), start with

f(x) = 6x+2

Take out all the x's and replace them all with empty parentheses (   )

f(    ) = 6(    )+2

Put 2x+1 inside the empty parentheses

f(2x+1) = 6(2x+1)+2

Simplify only the right side:

f(2x+1) = 12x+6+2

f(2x+1) = 12x+8

Edwin
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

I'm assuming f(x) is some linear function. If so, then,
f(x) = mx+b
m = slope
b = y intercept

Replace every x input with (2x+1) and simplify
f(x) = mx+b
f(2x+1) = m(2x+1) + b
f(2x+1) = m*2x+m*1 + b
f(2x+1) = 2mx+m+b

This is equal to 6x+2 since f(2x+1) = 6x+2 is given to us, so,
2mx+m+b = 6x+2
2mx+m+b = 6x+2

Equating the two separate pieces has us see that
2mx = 6x
2m = 6
m = 3

and also,
m+b = 2
3+b = 2 ... plug in m = 3
b = -1

---------------------------

Therefore, m = 3 and b = -1, meaning we can say
f(x) = mx+b
f(x) = 3x+(-1)
f(x) = 3x-1

note that replacing every x with (2x+1) leads to...
f(x) = 3x-1
f(2x+1) = 3(2x+1)-1
f(2x+1) = 3(2x)+3(1)-1
f(2x+1) = 6x+3-1
f(2x+1) = 6x+2
which helps confirm our answer.
Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.

            Right and  CANONICAL  way to do it is as follows. 


Let "t" be new variable, t = 2x+1.


Then  x = .


Express  6x+2  via  "t" :


    6x+2 =  = 3*(t-1) + 2 = 3t - 3 + 2 = 3t-1.


Thus  f(t) = 3t-1.


Here "t" is  a "mute" / ("silent") variable.  It can be replaced by ANY symbol in both sides simultaneously.


For example, it can be replaced by x.  So


    f(x) = 3x-1.      ANSWER

Done, completed and explained.


This my post is the  standard  MANTRA  to use for solution to each and every similar problem.


--------------

        //   Mantra is called "mantra",  because it  DOES  NOT  ALLOW  to change any word (!)



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