fonction of composition

Question

fonction of composition (Fog)(x)=1/x^2 and g(x)= 2x+1
find f(x)=?

Answer 1

What operation would have to be performed on g(x) to get just x?  ANSWER: ½(g(x)-1)

That gives you x. Now what operation do you have to do to that, to get 1/x²?  ANSWER: raise that to the -2 power.

 

So try this:  f(x) = [½(x-1)]^-2 ...combining the two necessary steps listed above.

 

It works! YOU VERIFY!

Answer 2

f(g(x)) = 1 / x²

 

f(2x + 1) = 1 / x²

 

f(2x + 1 - 1) = 1 / (x - 1)²

 

f(2x) = 1 / (x - 1)²

 

f(2x / 2) = 1 / ((1/2)(x - 1))²

 

f(x) = 1 / [(1/2)(x - 1)]²

 

 

check:

 

f(2x + 1) = 1 / [(1/2)(2x + 1 - 1)]²

 

              = 1 / x²

 

checks out!!