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

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?

• e.g., $y^2=x^2$ is a relationship but $y$ and $x$ are not a function of each other – J. W. Tanner Jun 20 '19 at 14:50
• For instance, $x^2+y^2=1$ is an equation whose solution set describes the unit circle. There is not however a function which describes the same set of values as it would not pass the "vertical line test" – JMoravitz Jun 20 '19 at 14:51
• Consider the equation of the circle : $x^2+y^2=1$. We cannot express it in the form : $y=f(x)$. – Mauro ALLEGRANZA Jun 20 '19 at 14:51
• – Ethan Bolker Jun 20 '19 at 14:53
• Besides the examples given, there are many cases where we have actually a function but no formula is known. Let $\pi(n)$ be the $n$th prime number, so $\pi(1)=2,\pi(2)=3,\dots$. $\pi$ is certainly a function from the positive integers to the positive integers, but so far as anyone knows, there is no formula for $\pi.$ – saulspatz Jun 20 '19 at 15:16

## 2 Answers

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.

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.