Question

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

Answer 1

For: `(f@g)(x)`
`h(x) = [3x]_(x=g(x)=1/x) = 3*1/x=3/x`

For: `(g@f)(x)`
`r(x) = [1/x]_(x=f(x)=3x) = 1/(3x`

Explanation:

Given:
`f(x) = 3x and g(y) =1/y`

Required:
Composite functions: `a) =>(f@g)(x) and b) =>(g@f)(x)`

Solution Strategy:
- Step 1: Rewrite the composition `h(x) = (f og)(x) =>f(g(x))`.
- Step 2: Replace each occurrence of x found in the outside function with the inside.
- Step 3: Simplify the answer

`--------------------`
For: `(f@g)(x)`

`color(crimson)(Step- 1)`
`h(x) = f(g(x))=f(1/x)`;
`color(crimson)(Step- 2)`
Replace each occurrence of `x` in `f(x)` with `g(x) = 1/x`
`h(x) = [3x]_(x=g(x)=1/x) = 3*1/x=3/x`
The dummy variable is not relevant so you can do this in terms of
`x or y or theta`a

`color(crimson)(Step- 3)` function in simplest form no step 3 needed

`--------------------`

For: `(g@f)(x)`

`color(fuchsia)(Step- 1)`
`r(x) = g(f(x))=g(3x)=`;
`color(fuchsia)(Step- 2)`
Replace each occurrence of `x` in `g(x)` with `f(x) = 3x`
`r(x) = [1/x]_(x=f(x)=3x) = 1/(3x`
`color(fuchsia)(Step- 3)` function in simplest form no step 3 needed