Expert: Paul Klarreich - 9/16/2009

Question
Find The Functions
f(g(x)), g(f(x)), f(f(x)), g(g(x)) and their domains.

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

Answer
Questioner: patty
Country: United States
Category: Calculus
Private: No
Subject: calculus
Question: Find The Functions
f(g(x)), g(f(x)), f(f(x)), g(g(x)) and their domains.

f(x)=2x^2-x
g(x)= 3x+2
.......................................
Hi, Patty,

When I get questions like this, I know it must be September.  

These are substitution-into-functions questions, and the basic method you MUST USE looks like this:

For  f(g(x)), write the pattern for f:

f(x) = 2x^2 - x

Put () for the argument.  (Here the argument is called x.)

f(  ) = 2(  )^2 -  (  )

Insert your whatever:

f(g(x)) = 2(g(x))^2 -  (g(x))

Substitute for the whatever on the right:

f(g(x)) = 2(3x + 2)^2 -  (3x + 2)

Now use the good algebra your high school teacher taught you and simplify.  Naturally, I leave that to you.

In this (and all of these) your domain is all reals, because these are polynomials.
.............................

For  f(f(x)), write the pattern for f:

f(x) = 2x^2 - x

Put () for the argument.  (Here the argument is called x.)

f(  ) = 2(  )^2 -  (  )

Insert your whatever:

f(f(x)) = 2(f(x))^2 -  (f(x))

Substitute for the whatever on the right:

f(g(x)) = 2(2x^2 - x)^2 - (2x^2 - x)

etc.
...............................
For  g(f(x)), write the pattern for g:

g(x)= 3x + 2

g() = 3() + 2

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

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

etc.
...............................
For  g(g(x)), write the pattern for g:

g(x)= 3x + 2

g() = 3() + 2

g(g(x)) = 3(g(x)) + 2

g(g(x)) = 3(3x + 2) + 2

etc.