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

Apr 20, 2017

See below

#### Explanation:

The domain is $\mathbb{R}$ since the relation is well defined for all $x \in \mathbb{R}$.
The codomain is also $\mathbb{R}$ since for every ${y}_{0} \in \mathbb{R}$ we can take ${x}_{0} = \frac{1}{3} {y}_{0} + \frac{4}{3}$ and get that

$3 {x}_{0} - 4 = 3 \left(\frac{1}{3} {y}_{0} + \frac{4}{3}\right) - 4 = {y}_{0}$

The relation is a function:

EXISTENCE:

for every $x$ in the domain ($\mathbb{R}$) there is $y$ in the codomain ($\mathbb{R}$) such that $y = 3 x + 1$: trivial, just take $y = 3 x + 1$.

UNIQUENESS:

If ${y}_{1}$ and ${y}_{2}$ are images of the same ${x}_{0}$, then

${y}_{1} - {y}_{2} = 3 {x}_{0} + 4 - 3 {x}_{0} - 4 = 0 R i g h t a r r o w {y}_{1} = {y}_{2}$