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The definition of a function is "A function is a relation in which there is never more then one value of the dependent variable for every value of the independent variable."

Since a quadratic equation has two solutions for every input does this mean that the quadratic is not a function?

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You need to be more specific about the "quadratic equation" and what its "input" means. Not all quadratic equations have two solutions, and the notion of dependent vs. independent variable would not be self-evident from looking at a quadratic equation in two variables. –  hardmath Apr 29 '13 at 14:31
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The short answer: no equation is a function. Is the equation $3x=5$ a "function?" In what sense? –  Thomas Andrews Apr 29 '13 at 14:48
    
As several have pointed out, your "equation" is a bit to vague to be clear. Now, such an equation might imply a function. Perhaps, we could write down specific assumptions under which this works, wrap it all up into a theorem and call it "the implicit function theorem". –  Mark McClure Apr 29 '13 at 14:52
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Thomas: It could be viewed as a function from some domain of values for $x$ to the set $\{$True, False$\}$. –  Hammerite Apr 29 '13 at 15:51
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6 Answers 6

I think you're getting your dependent and independent variables mixed up. Given a quadratic equation, say $y=ax^2+bx+c$, the independent variable is $x$, whereas the dependent variable is $y$. Quadratics have at most two solutions for every output (dependent variable), but each input (independent variable) only gives one value.

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The function $f(x)=ax^2+bx+c$ is a quadratic function.

Now, if you try to solve a quadratic equation, you get often two solutions, but this is not the same as calculating the function. What does this actually shows is that the quadratic function takes many values twice, and in particular doesn't have an inverse.

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You are mixing up "function" ($f \colon \mathcal{A} \rightarrow \mathcal{B}$, where $f$ assigns a unique value in $\mathcal{B}$ to each $x$ in $\mathcal{A}$, written $x \mapsto f(x)$; with "equation", a question of the form $f(x) = 0$ where the values (if any) of $x$ that satisfy it are sought. An equation might have no, one, or several solutions. For example, if you look for $x \in \mathbb{R}$, you have equations $x^2 + 1 = 0$ (no solutions), $x^2 = 0$ (one solution), $x^2 - 4 = 0$ (two solutions) and $0 \cdot x = 0$ (infinite number of solutions).

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Consider an equation $ax^2+bx+c=0$. Let $R(a,b,c)=\{x_1,x_2\}$ be the set of its roots. Then the set $R(a,b,c)$ is a function of $a,b,c$. However, every function must take only one value so there is no function that takes both values $x_1$ and $x_2$ (according to your definition).

On the other hand, quadratic polynomial $f(x) = ax^2 +b x +c$ is a function. However, $f(x)$ is not the root of the equation $ax^2+bx+c=0$. So the fact that $ax^2+bx+c=0$ has two roots is irrelevant.

Finally, the sentence "a quadratic equation is a function" is very ambiguous, to say the least; it's not clear what it means.

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No. Using your language (which I wouldn't consider very proper, see the other answers), your two solutions are choices of the independent variable. The fact that there are two solutions to most quadratic equations $ax^2+bx+c=0$ implies that the function $f(x)=zx^2+bx+c$ is not injective. But it is still a function: for every choice of $x$, there is a well-defined choice for $f(x)$.

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Equations are usually not functions (see below). An equation is a logical statement, and as such it is either true or false. Specifically, it is a statement which asserts that two mathematical expressions denote objects or quantities which are equal: equation is the situation of of making two equal. The equation has a close relative, the inequality, which is a statement that one quantity is greater than or less than another.

An expression, or formula, however can be a function, if we make it clear which variables in that formula are to serve as parameters. For instance, the quadratic formula $x^2 + 2x + 1$ will serve us as a function of $x$. There is one variable in it, so if it is going to be a function, that variable is it.

If we are confronted with a formula with multiple symbols, like $ax^2 + bx + c$ then it is not clear how it is intended to be used as a function. Some of the symbols might be considered to be held constant. Usually something which looks like $ax^2 + bx + c$ is a function of $x$, and the $a, b, c$ are coefficients, but that is not necessarily so; we are prejudiced toward interpreting it that way because it resembles a common convention. It could easily be a function of three variables $f(a, b, c) = ax^2 + bx + c$, where $x$ is a constant, or a free variable which parametrizes the function.

This is why the function notation is useful: it makes it clear which symbols are bound as parameters and which are free. But in some situations we can drop the notation, where it is clear which variables are which, and just work with formulas.


Can equations be functions? Since an equation is itself an expression made up of symbols and operators, and has a value (boolean: true or false), it can be regarded as a function, where some or all of the symbols are considered independent variables.

This is in fact what happens when we do not know whether or not the equation is true. When we solve an equation, we find combinations of variables which make the equation true. Other combinations of variables exist which make it false.

For instance, take the equation $x^2 + y^2 = r^2$. Let us hold $r$ constant. We can regard the equation to be a boolean function: $f(x,y) := x^2 + y^2 = r^2$. Note that I used a different equal sign $:=$ to denote definition, because we now have an equal sign that can appear in a function body.

This function $f(x,y)$ is true whenever $x$ and $y$ lie on the circle of radius $r$ around the origin, and false for all other values.

The visualization software GrafEq takes this perspective on equations in order to plot their graphs.


When we consider a two-variable equation or inequality to be a boolean function, we can imagine that black ink is plotted in the 2D space for those coordinates where the inequality or equation is true, and white paper remains everywhere.

Now what if we can find an inequality which, when plotted this way, reproduces an image of its own notation? Such a formula is known is known as Tupper's self-referential formula, a form of Quine.

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