Let F(x)=3x and g(y)=1y, how do you find each of the compositions and domain and...

For: ${\displaystyle (f\circ g)\left(x\right)}$
${\displaystyle h\left(x\right)={\left[3x\right]}_{x=g\left(x\right)=\frac{1}{x}}=3\cdot \frac{1}{x}=\frac{3}{x}}$
For: ${\displaystyle (g\circ f)\left(x\right)}$
${\displaystyle r\left(x\right)={\left[\frac{1}{x}\right]}_{x=f\left(x\right)=3x}=\frac{1}{3x}}$

Required:
Composite functions: a) ${\displaystyle \Rightarrow (f\circ g)\left(x\right)}$ and b) ${\displaystyle \Rightarrow (g\circ f)\left(x\right)}$
For: ${\displaystyle (f\circ g)\left(x\right)}$
Step 1
${\displaystyle h\left(x\right)=f\left(g\left(x\right)\right)=f\left(\frac{1}{x}\right)}$
Step 2
Replace each occurrence of x in ${\displaystyle f\left(x\right)}$ with ${\displaystyle g\left(x\right)=\frac{1}{x}}$
${\displaystyle h\left(x\right)={\left[3x\right]}_{x=g\left(x\right)=\frac{1}{x}}=3\cdot \frac{1}{x}=\frac{3}{x}}$
The dummy variable is not relevant so you can do this in terms of x or y or ${\displaystyle \theta }$
Step 3
function in simplest form no step 3 needed
For: ${\displaystyle (g\circ f)\left(x\right)}$
Step 1
${\displaystyle r\left(x\right)=g\left(f\left(x\right)\right)=g\left(3x\right)=}$
Step 2
Replace each occurrence of x in ${\displaystyle g\left(x\right)}$ with ${\displaystyle f\left(x\right)=3x}$
${\displaystyle r\left(x\right)={\left[\frac{1}{x}\right]}_{x=f\left(x\right)=3x}=\frac{1}{3x}}$
Step 3
function in simplest form no step 3 needed