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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?

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  • $\begingroup$ e.g., $y^2=x^2$ is a relationship but $y$ and $x$ are not a function of each other $\endgroup$ – J. W. Tanner Jun 20 '19 at 14:50
  • $\begingroup$ 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" $\endgroup$ – JMoravitz Jun 20 '19 at 14:51
  • $\begingroup$ Consider the equation of the circle : $x^2+y^2=1$. We cannot express it in the form : $y=f(x)$. $\endgroup$ – Mauro ALLEGRANZA Jun 20 '19 at 14:51
  • $\begingroup$ Related: math.stackexchange.com/questions/2738360/… $\endgroup$ – Ethan Bolker Jun 20 '19 at 14:53
  • $\begingroup$ 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 $\pi.$ $\endgroup$ – saulspatz Jun 20 '19 at 15:16
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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.

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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.

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