What does it mean to define something as a curve?

Question

What does it mean to define something as a curve?

Does writing $y=x$ means I'm explicitly talking about a curve and writing $f(x)=x$ means I'm defining it as a function?

Answer 1

Both of the writings are equivalent.

$$f(x) = x$$

means that you're defining a function $f$ whose only variable is $x$, hence you define a function of a single variable which is $x$ (either real or complex, this is up to you).

Writing

$$y = x$$

Is a sort of "geometrical" way to define the same function as a curve: in the cartesian plane this curve will be a straight line, a segment, the bisection of the first and third Cartesian plane quadrant.

There is nothing strange to define something as a curve, as long as you can represent it in the Cartesian plane. It's like to define a parabola $x^2$ or a random curve $\ln(x) + x^2$.

They are both a curve and a function. It depends on what you need to do.

Answer 2

When you write $y=x$ it is implicitly assumed that you are looking at points $(x,y)$ in Cartesian plane that satisfy the given equation. More formally, you are looking at set $\{(x,y)\in\mathbb R^2\mid y = x\}$. It turns out that this is the graph of the function $f\colon\mathbb R\to\mathbb R$ given by formula $f(x) = x$.

So, what exactly is the difference between the two?

For one, it can be dangerous not to distinguish between a function and its graph. For example, you could consider functions $f\colon\mathbb R\to\mathbb R$, $g\colon\mathbb R\to\mathbb R_{\geq 0}$, $h\colon\mathbb R_{\geq 0}\to\mathbb R$ and $k\colon \mathbb R_{\geq 0} \to \mathbb R_{\geq 0}$ all defined by the same formula: $f(x) = x^2$, $g(x) = x^2$, $h(x) = x^2$, $k(x) = x^2$, i.e. their graphs all satisfy the same equation $y = x^2$. But, $k$ is bijective, $h$ is injective, but not surjective, $g$ is surjective, but not injective and finally, $f$ is neither.

On the other hand, there are equations whose set of solutions is not a graph of any function, for example, $y^2 = x$ defines parabola "parallel" to $x$-axis, while neither $f(x) = \sqrt x$ nor $g(x)=-\sqrt x$ (nor infinitely many more discontinuous functions) will fully encapsulate. This is because for any $x>0$ there are two possible $y$'s that satisfy the equation $y^2 = x$ and this relation cannot be function by definition.

Finally, equations can be solved over sets other than $\mathbb R$. So, really, without context, these can be entirely different beasts.