What does it mean when not every relationship expressed by a equation can also be expressed as a function with a formula?

Question

I'm studying a pre-calculus textbook and it mentioned this:

"It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula."

Can someone help me understand this by giving examples?

Answer 1

Example: circle. If a point in the plane is a distance of $1$ away from the origin, then there is a relationship between that point's $x$ coordinate and its $y$ coordinate. That relationship is given by the equation $$ x^2 + y^2 = 1 $$ However, it is impossible to write this relationship as $y = f(x)$ (or as $x = f(y)$) for a function $f$, as for any given $x$ there may be several $y$ which fulfill the relationship, and functions cannot return several values at once, by definition.

Answer 2

This depends on how you define 'function'.

If you are working off the set-theoretic [which is the standard] definition of a function, then a function is a set of points $(x,y)$ such that for each $x$, there is one and only one $y$.

Thus if $y^2=x$, then the equation $f(x)=y$ does not represent a function, since $y=\sqrt{x}$ and $y=-\sqrt{x}$ are both valid solutions.

A relation, on the other hand may be any set of points.

There are specific contexts in which it makes sense to use a different definition of 'function', such as complex analysis and certain theories of computation. In these cases almost any equation can be thought of as a function, but you tend to lose the requirement that each input yields only a single output. For example, the complex multivalued function $f(z)=\pm\sqrt{z}$ has two output values for each complex number $z$.

If you prefer to think of functions as a black box process, and are familiar with the concept of 'images', then an expression $f$ is a function iff for every singleton set $X$, $f(X)$ is also a singleton set.

In the case of $y^2=x$, we know that the expression $f(x)=y$ does not represent a function because, for example, $f(\{4\})=\{2,-2\}$.

Most equations for implicit curves are not functions, and many examples can be found in this list.