Question

How do you find the inverse of `f(x)= 1/3x + 2` and is it a function?

Answer 1

`color(blue)(g(x)=3x-6)` is a function
(which is the inverse of `f(x)=1/3x+2`)

Explanation:

If `g(x)` is the inverse of `f(x)`
then by definition
`color(white)("XXX")g(f(x))=x`
and
`color(white)("XXX")f(g(x))=x`

Since `f(color(red)(x))=1/3color(red)(x)+2`
then
`color(white)("XXX")f(color(red)(g(x)))=1/3color(red)(g(x))+2`

But from our original definition we know that
`color(white)("XXX")f(g(x))=x`

Therefore
`color(white)("XXX")f(g(x))=1/3g(x)+2=x`

`color(white)("XXX")rArr1/3g(x)=x-2`

`color(white)("XXX")rArrg(x)=3x-6`

Since any value of `x` gives a single value for `g(x)`
it follows that `g(x)` is a function.