Difference Between an Equation and Function?

[pandaexpress]

So, I'm relearning some Algebra for my Calculus class and I wanted to ask what is the difference between an equation and a function? For example,

y = 3x + 2
f(x) = 3x + 2

Both are technically equations right? Why not just write them one way or the other way?

Thanks!

 

[Geofleur]

Two differences immediately come to mind:

(1) A function represents a different point of view from an equation. A function acts on something and produces something. In $f(x) = 3x + 2$, for example, the function $f$ acts on $x$ and produces $3x + 2$. Also, we are invited to think of $f$ as an independent entity that could be expressed in terms of different coordinates. If $x$ and $y$ are Cartesian coordinates, for example, the same function in polar coordinates (where $x = r\cos \theta$ and $y = r \sin \theta$) would read $f(r,\theta) = 3r\cos \theta + 2$. Sometimes looking at the same thing from a different point of view makes the difference between being able to solve a problem and not.

(2) An equation can represent many different functions. For example, $y = 3x + 2$ could represent $f(x) = 3x + 2$, but it could also represent $g(y) = (y-2)/3$. Leaving the relationship as an equation leaves open the question of which function you may want.

 

[aikismos]

To add to Geofleur's points, the point of stating an equation in functional notation adds some extra information when stating the equivalence relation (in this case an arithmetic equality). The moment you declare an equation a function, you're talking about a relationship between the domain and the codomain such that every input only produces one output (in a single-variable function). A parabola whose axis of symmetry is vertical has one output for every input, even if multiple inputs produce the same output: $f(x) = x^2 \rightarrow f(2) = f(-2) = 4; \{(2,4), (-2,4)\}$. Once you turn the axis of symmetry horizontal $f^{-1}(x) = \sqrt{x} \rightarrow f(4) =\pm 2; \{(4,2), (4,-2)\}$ then the mapping of inputs to outputs becomes
ambiguous. That's why we typically restrict codomains of inverse functions. In the listed example, generally we graph the principal (or positive) square root only.

Why is all of this important? Because often the mappings between domains and codomains in functions can allow us to draw conclusions about functions and their inverses which allow us to extend our mathematical theorems and statements in a meaningful way. The study of functions is rather crucial to understanding math theories much more broadly than the differential and integral calculi. (See https://en.wikipedia.org/wiki/Function_(mathematics).)

 

[Mark44]

To add to Geofleur's points, the point of stating an equation in functional notation adds some extra information when stating the equivalence relation (in this case an arithmetic equality). The moment you declare an equation a function, you're talking about a relationship between the domain and the codomain such that every input only produces one output (in a single-variable function). A parabola whose axis of symmetry is vertical has one output for every input, even if multiple inputs produce the same output: $f(x) = x^2 \rightarrow f(2) = f(-2) = 4; \{(2,4), (-2,4)\}$. Once you turn the axis of symmetry horizontal $f^{-1}(x) = \sqrt{x} \rightarrow f(4) =\pm 2; \{(4,2), (4,-2)\}$ then the mapping of inputs to outputs becomes
ambiguous.

BUT, the symbol $\sqrt{4}$ represents the positive square root, or 2 in this case.
Also, you have a typo with f(4), which I'm sure you meant as $f^{-1}(4)$. In any case, because f as you defined it isn't one-to-one, its inverse is not a function.

That's why we typically restrict codomains of inverse functions. In the listed example, generally we graph the principal (or positive) square root only.

Why is all of this important? Because often the mappings between domains and codomains in functions can allow us to draw conclusions about functions and their inverses which allow us to extend our mathematical theorems and statements in a meaningful way. The study of functions is rather crucial to understanding math theories much more broadly than the differential and integral calculi. (See https://en.wikipedia.org/wiki/Function_(mathematics).)

 

[fresh_42]

A function $f : X → Y$ is a special relation, i.e. a subset $M$ of $X \times Y$, such that {$(x , y_1) , (x , y_2)$} $∈ M$ implies $y_1 = y_2$.

A equality is a equivalence relation ~ (reflexive, symmetric, transitive) on $X$, i.e. the quotient set $X /$~ such that $x_1 = x_2$ if and only if { $x_1$ } ~ { $x_2$ }.

 

[micromass]

{$(x , y_1)$} $=${$(x , y_2)$}

This notation makes no sense to me. Why use { } here?

 

[Svein]

Both are technically equations right? Why not just write them one way or the other way?

Wrong.

• f(x) = 3x + 2 defines a function, not an equation.
• y = f(x) assigns the value of the function f(x) to the variable y.
• For what values of x is 3x + 2 = 17 is an equation. It may or may not have a solution.

 

[jbriggs444]

f(x) = 3x + 2 defines a function, not an equation

While function definitions often include a particular style of equation, that style of equation does not always amount to a function definition. For instance:

Let f be the function defined by f(x) = sin x. What values of x satisfy f(x) = 3x + 2 ?

 

[aikismos]

Wrong.

• f(x) = 3x + 2 defines a function, not an equation.

Wouldn't be better to say that f(x) = 3x + 2 defines a function, not a relation? In math, if it has an equal sign, as far as I understand, it's an equation.

 

[fresh_42]

I mean one lives in a product space, the other in a quotient space. It couldn't be more different!