Is x=y=z a function?

Question

Is x=y=z a function?

I was trying to graph things like x=y=z and x=2z, y=3z, z=z earlier today

and noticed that 3d graphers can't graph these.

Is this because these are not functions but mere relations?

Would they be a straight line in a 3d-space if I were to graph them?

Thank you for reading. Any comments would help very much.

Answer 1

Is $x=y=z$ a function?

What you write here is usually considered as a compact notation for a system of two equations: $$\left\{\begin{align} x = y \\ y = z \end{align}\right.$$ Both equations represent a plane in three-dimensional space and the system thus represents the points common to both planes. If the planes are non-parallel, they intersect in a line.

Another way to see this would be to solve the system; setting $z=t$ easily yields $x=y=t$ too and the solution has the following parametric form: $$\left\{ \left(t,t,t\right) \vert t \in \mathbb{R} \right\}$$ which is a parametric representation of a line, through the origin and with $(1,1,1)$ as a direction vector. Most software can probably plot it in this form.

You could do the same for the second example you gave, it is another line.

Answer 2

I guess that you confuse function (a relation between two sets) and graph of a function (a curve), and your actual question is "is the plot of the set defined by $x=y=z$ a straight line?". The answer is yes (and these equations are not "a function").

Usually, graphers are able to plot 3D curves given their parametric equations, i.e. of the form

$$\begin{cases}x=f(t),\\y=g(t),\\z=h(t)\end{cases}$$ or in vector form

$$\vec t=\vec f(t)$$ where this time $\vec f$ is a function.

For your case, you can very well specify a parametric equation of the form

$$\begin{cases}x=t,\\y=t,\\z=t\end{cases}$$ which fulfills the initial system.

Answer 3

$x=y=z$ is shorthand for the function

$f : \mathbb{R} \rightarrow \mathbb{R}^3 : t \mapsto (t,t,t)$

Answer 4

It represent a line passing the origin which the equation of its projections(a line passing the origin) in x,y plane is $y=x$ and in y,z plane is $y=z$ and finally in x,z plane is $x=z$. So instead of writing:

$y=x$ , $y=z$, $x=z$ we compact it and write:$y=x=z$