Question #a0d9a

Nov 21, 2017

Yes, this is a function. Domain: $x \in \mathbb{R}$. Range: $y \in \mathbb{R}$

Explanation:

A function can be thought of as a "rule" to follow to get an output. The most important thing about a function is that for a set input, there is only one output. This is what makes $y = {x}^{2}$ a function, but ${x}^{2} + {y}^{2} = 9$ is not. Take $f \left(x\right) = {x}^{2}$. For a set value of x, say $x = 4$, we get a single output, $f \left(x\right) = 16$
(As a side note, this does not mean that one output cannot have multiple inputs. For example, if $f \left(x\right) = {x}^{2}$ and $f \left(x\right) = 16$, then $x = \pm 4$. Despite this, either of these as inputs only gives one answer, so this is still a function.)

The equation given in this question, $3 x + y = 6$, is the equation of a straight line. This means that for any given x value, there is only one y value. Indeed, this can also be written as $f \left(x\right) = 6 - 3 x$, further showing it is a function.

Additionally, this will hold for all values of x. There are no asymptotes, or imaginary parts of the graph. This means that the domain is all the real numbers of x, often written $x \in \mathbb{R}$, "x is a member of the set of real numbers".
The graph also has no maximum or minimum. The graph will extend to positive and negative infinity on the y axis; also think of this that any y value will give a corresponding x value. So the range is $y \in \mathbb{R}$, "y is a member of the set of real numbers"