Question

How do you find the domain and range and determine whether the relation is a function given :`y=3x-4`?

Answer 1

See below

Explanation:

The domain is `RR` since the relation is well defined for all `x in RR`.
The codomain is also `RR` since for every `y_0 in RR` we can take `x_0=1/3y_0+4/3` and get that

`3x_0-4=3(1/3y_0+4/3)-4=y_0`

The relation is a function:

EXISTENCE:

for every `x` in the domain (`RR`) there is `y` in the codomain (`RR`) such that `y=3x+1`: trivial, just take `y=3x+1`.

UNIQUENESS:

If `y_1` and `y_2` are images of the same `x_0`, then

`y_1 - y_2=3x_0+4 - 3x_0-4=0 Rightarrow y_1 = y_2`