Give a serious explanation of the difference between an equation and a function.

What's the difference between an equation and a function? I mean, I am not seeking for a high school-like answer like "an equation has an equals sign". I want to know what is the fundamental difference between the two. Why we need these two concepts, that seems to me to be the same thing? In physics we heard a lot about equations, but if we call all them functions, wouldn't we be right?

Solution 1:

They are the same in a sense. A function is technically an "equation", but an equation is not necessarily a function. It is incontrovertibly true; however, to say that any two values separated by an equals sign is considered an equation. $5=2+3$ is an equation, but not a function. $y=2x+3$ or equivalently, $f(x)=2x+3$ is a function, but also an equation because it involves an equals sign. Functions are equations that have both constants and variables which arrange in some way to map the function's value about those variables' dimensions.

Solution 2:

A function is something that yields a well-defined output for any appropriate input. Some, but not all, functions can be represented by equations, and not every equation naturally represents a function. In order to see an equation as a function, you have to decide what the output (the dependent variable) is supposed to be, and then you have to determine whether that has a unique value, given inputs that fill in the rest of the blanks in the equation.

Consider the equation $w=z^2$. We could interpret this equation as giving us a formula for $w$, as a function of $z$, because for a given $z$ value, we can say exactly what $w$ must be.

On the other hand, that same equation doesn't give us a formula for $z$ as a function of $w$. That's because, using the value $w=1$, it's impossible to say whether our output should be $z=1$ or $z=-1$.

I hope that this answer addresses your question appropriately.

Solution 3:

In many cases, the distinction is subtle. A function is formally defined as a mapping between two sets $X$ and $Y$ in which each element of $X$ is assigned (that is, maps to) an element in $Y$. An equation is a statement about the equivalent value of two expressions. In this sense, a function $f$ is an equation because the symbol $f(x)$ is equivalent to some expression, probably involving $x$. A subtlety is that while we often express a function as $f(x)=...$, this is an equation while the function is more properly denoted $f$ (without reference of $x$). An example (from physics, for your sake) which would not (I think) be appropriately called a function is $$\nabla\cdot E=\frac{\rho}{\varepsilon_0}.$$ This is one of Maxwell's equations which is certainly not (as written) a mapping between sets.