Difference Between an Equation and Function?

In summary, the difference between an equation and a function is that a function represents a different point of view and acts on something to produce something, while an equation can represent many different functions. The notation of functional notation adds extra information when stating an equivalence relation and is important in understanding mathematical theories. It is incorrect to refer to a function definition as an equation.
  • #1
pandaexpress
14
0
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!
 
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  • #2
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.
 
  • #3
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).)
 
  • #4
aikismos said:
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.
aikismos said:
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).)
 
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  • #5
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 ## }.
 
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  • #6
fresh_42 said:
{##(x , y_1)##} ## = ##{##(x , y_2)##}

This notation makes no sense to me. Why use { } here?
 
  • #7
pandaexpress said:
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.
 
  • #8
Svein said:
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 ?
 
  • #9
Svein said:
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.
 
  • #10
I mean one lives in a product space, the other in a quotient space. It couldn't be more different!
 
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What is the difference between an equation and a function?

An equation is a mathematical statement that shows the relationship between two or more variables, often expressed using symbols and operations. A function, on the other hand, is a mathematical rule that assigns exactly one output value for each input value. In other words, a function is a type of equation that describes a specific relationship between input and output values.

Can an equation and a function be used interchangeably?

No, an equation and a function are not interchangeable. While all functions are equations, not all equations are functions. For an equation to be considered a function, it must pass the vertical line test, meaning that a vertical line cannot intersect the graph of the equation more than once.

How do you represent an equation and a function?

An equation can be represented in various ways, such as using symbols, words, or graphs. A function is commonly represented using function notation, where the input value is placed inside parentheses and the output value is written after the function name, such as f(x) = 2x + 1.

What are the similarities between an equation and a function?

Both equations and functions involve mathematical relationships between variables. They can also both be used to solve problems and make predictions. Additionally, both equations and functions can be graphed on a coordinate plane.

What are the real-life applications of equations and functions?

Equations and functions are used in various fields, including physics, engineering, economics, and biology. They can be used to model and understand real-world phenomena, make predictions, and solve problems. For example, in physics, equations and functions can be used to describe the relationship between distance, time, and velocity.

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