Answer
$ a) (f\circ g)(x)=6x+3,\ Domain: \mathbb{R} \\ b) \ (g\circ f)(x)=6x+9,\ Domain: \mathbb{R} \\ c) (f\circ f)(x)=4x+9, Domain: \mathbb{R} \\ d) (g\circ g)(x)=9x,\ Domain: \mathbb{R}$
Work Step by Step
We are given:
$ f(x)=2x+3$ and $ g(x)=3x ; \ Domain: \mathbb{R}$
We find the composite functions as follows:
$a) f\circ g(x)=f[g(x)]=2g(x)+3 \\=2(3x)+3 \\=6x+3 ; Domain: \mathbb{R}$
$b) (g\circ f)(x)=g[f(x)] =3f(x) \\=3(2x+3) \\=6x+9 ; \ Domain: \mathbb{R}$
$c) f\circ f(x)=f[f(x)]=2f(x)+3 \\=2(2x+3)+3 \\=4x+6+3 \\ =4x+9; \ Domain: \mathbb{R}$
$d) (g\circ g)(x)=g[g(x)] =3g(x) \\=3(3x) \\=9x \ Domain: \mathbb{R}$
