use the given pair of functions to find and simplify express

use the given pair of functions to find and simplify expressions for the following functions and state the domain of each using interval notation.

(gf)(x)     (f g)(x)      (f f)(x)

f(x)= 3x2, g(x)= (x+1)

Solution

f(x)= 3x^2, g(x)= (x+1)

i) ( g o f)(x) ; plug x = f(x) in g(x)

( g o f)(x) = sqrt(3 -x^2 +1) = sqrt( 4-x^2)

Domain : 4-x^2 > =0

[ -2 , 2 ]

ii) ( f o g)(x) plug x = g(x) in f(x)

( f o g)(x) = 3 - {sqrt( x+1) }^2 = 3 - (x+1) = 2- x

Domain : ( - inf , inf)

iii) (f o f)(x) ; plug x = f(x) in f(x)

(f o f)(x) = 3 -( 3- x^2) = x^2

Domain : ( - inf, inf)