Let's say I have an equation $y = mx + b$. The value $b$ is commonly called a "constant" because the value does not change for any input $x$.
Now let's say I have another equation $f$. This equation does not change for any scenario it is being used in. Can I say that this equation is "constant"? Or is there another term for describing this equation?
Please let me know if I should clarify my question. Thank you so much!
A function whose value does not depend on its inputs, for example $f(x) = 42$, is indeed called a constant function (or just "constant").
It doesn't really make sense to call an equation constant. An equation which is always true regardless of the values of its free variables is called a tautology. If an equation sets a variable to a constant (for instance, $y=42$), we would say that $y$ is constant, but we wouldn't call the equation itself constant.
A short answer to your question is: Yes, although depends on the definition of functions/values but read on.
I shall refer to constants or rather quantities which do not change as scalars for now. Scalars are quantities that are fixed. Suppose a vector which is a list of numbers or a line originating from the origin that has magnitude and direction. Now, let $\mathbb{\mathcal{v}}$ be a vector with elements $(2,3)$, then what is $3\mathbb{\mathcal{v}}$?
Note that $\mathbb{\mathcal{v}}$ is just an arrow/line segment from origin towards $(2,3)$, and $3.\mathbb{\mathcal{v}}$ is nothing but an extension of this line segment to $3.(2,3)=(6,9)$, so basically $\mathbb{\mathcal{v}}$ is "scaled" three times to become $3\mathbb{\mathcal{v}}$. Hence, the number $3$ is called a scalar, because it scales.
Now in case of just the real number line, think of $\mathbb{\mathcal{v}}$ as $(\mathcal{\hat{i}},0)$ then, $3$ is nothing but $3.(1,0)$, which is scaling the unit vector thrice in the direction of positive $x$. Hence, $3$ is a scalar. Similarly, it can be said for any real number.
A key property of these scalars is that they do not undergo any transformation in any arithmetic operation, that is their value is fixed. For instance $-1.(2,0)$ scaled the vector twice in negative $x$ direction but no change to the value $-1$ occurred. It remained constant under linear transformation. And this holds for any other transformation involving scalars. Hence, the name "constant".
Now coming to your question, "Can a function be a constant?" What are functions? They are not "values" but just maps. A function $f(x)$ can be constant if no matter what the value of $x$ is, the value of $f(x)$ remains same. So, a function like $f(x)=4, \; \forall x \in \mathbb{R}$ is a constant function.
So, if I denote $f(x)$ by $y$, then I can say that in $y=4$, y is a constant and this in a very vague sense is a constant equation but it is better to avoid this terminology.
