Question

Let `f(x) =-2x+7` and `g(x) = -6x+3`. What is `f *g` and what is its domain?

Answer 1

Either `color(red)(12x+1)` or `color(blue)(12x^2-48x+21)`
depending upon interpretation of `f*g`

In either case the Domain is all Real values (`RR`)

Explanation:

Given
`color(white)("XXX")f(x)=-2x+7`
and
`color(white)("XXX")g(x)=-6x+3`

Possibility 1
If `f*g` is intended to mean `f@g` (i.e. `f(g(x))`)
then
`color(white)("XXX")(f(g(x)) = -2(g(x))+7`
`color(white)("XXXXXXXX")=-2(-6x+3)+7`
`color(white)("XXXXXXXX")=12x-6+7`
`color(white)("XXXXXXXX")=12x+1`
which is defined for all Real values of `x`
`rarr` Domain `=RR`

Possibility 2
If `f*g` is intended to mean `f xx g`
then
`color(white)("XXX")f(x)xxg(x)=(-2x+7)xx(-6x+3)`
`color(white)("XXXXXXXXXX")= 12x^2-48x+21`
which is defined for all Real values of `x`
`rarr` Domain `=RR`