#FunctionSimplification #GraphGeneration #Mathematics #GraphTheory #FunctionGraphs
Have you ever wondered if two different functions can lead to the same graph, but cannot be simplified to the same form? Let’s dive into this fascinating topic and discover the relationship between function simplification and graph generation. 📊
##Understanding Functions and Graphs
###What are Functions?
A function is a rule that assigns to each input value exactly one output value. In mathematical terms, a function f(x) takes an input value x and determines a unique output value f(x).
###What are Graphs?
In mathematics, a graph is a visual representation of the relationship between different variables. It consists of points called vertices and lines called edges, which connect the vertices.
##Simplifying Functions
When we talk about simplifying functions, we are referring to the process of expressing a function in a more concise and easily understandable form. Simplifying a function can involve various techniques, such as factoring, combining like terms, and using trigonometric identities.
##Generating Graphs from Functions
The process of generating graphs from functions involves plotting the input and output values of a function on a coordinate plane. This allows us to visually see the relationship between the input and output values, and how they change relative to each other.
##Can Two Different Functions Generate the Same Graph?
###Example 1: Linear vs Constant Functions
Let’s consider the functions f(x) = 2x and g(x) = 2. Both of these functions generate a horizontal line on the graph, where every input value x corresponds to an output value of 2. While these two functions have different forms, they produce the same graph – a horizontal line at y = 2.
###Example 2: Quadratic Functions
Next, let’s look at the functions f(x) = x^2 and g(x) = |x|. The function f(x) = x^2 generates a parabola that opens upwards, while the function g(x) = |x| produces a V-shaped graph. Despite their different forms, these two functions result in the same graph for non-negative values of x.
##The Concept of Equivalent Functions
In mathematics, two functions are considered equivalent if they produce the same output values for the same input values. This means that when two functions are graphed, they will overlap and produce the same set of points on the coordinate plane.
##Challenges in Identifying Equivalent Functions
While it is possible for two different functions to generate the same graph, identifying these equivalent functions can be a challenging task. In some cases, it may require advanced mathematical techniques or even the use of software to verify the equivalence of functions.
###Number of Intersection Points
One way to determine if two functions generate the same graph is to analyze the number of intersection points they have. If two functions intersect at the same set of points, it is an indication that they might be equivalent.
###Behavior at Critical Points
Another approach is to examine the behavior of the functions at critical points, such as maximum and minimum values. If the functions exhibit similar behavior at these points, it can suggest their equivalence.
##Conclusion
In conclusion, it is indeed possible for two different functions to generate the same graph, even if they cannot be simplified to the same form. The concept of equivalent functions opens up a world of possibilities in mathematics and graph theory, and challenges us to explore the intricate relationships between different mathematical entities.
So, the next time you encounter two different functions that seem unrelated at first glance, remember that they might just be different expressions of the same underlying relationship, and their graphs might align in surprising ways. Happy graphing! 🌐✨
No, at least if you use certain technical definitions.
In math speak, the “graph” of a function is the set of pairs (x, f(x)) for all x in the domain of f (recall: domain is the set on which the function is defined). The picture that you get is just putting points on paper at those coordinates.
So if you have two functions f and g whose graphs are identical, that means that they have the same domain, and that {(x, f(x)) for x in domain} = {(x, g(x)) for x in domain}.
In particular, this means that f(x) = g(x) for all x on which they are defined.
So far so good. But now the annoyingly pedantic part: “simplifying” just means replacing parts of an equation with something identical. Since f and g are identical, you can simplify by replacing one with the other by definition.
That’s not very satisfying though, and probably also isn’t what you meant. To technical up your question to mean what you meant, we’d probably have to rephrase:
Is it possible that there are two ways of defining a function, (ie as the sets of solutions to some equations) such that they result in the same function but that it cannot be proven that they result in the same function? This is closer to the feeling of your question which is “they’re really the same, but there’s no way to show that they are the same”.
I don’t know the answer to this, off hand, but if I had to guess, I’d say that the answer is probably yes. There are results from logic stating that any model of axioms meeting some set of restrictions must contain statements that are true (satisfied by the model) but aren’t provable from the axioms. It has been some time, but I would be kind of surprised if the proof could not be modified to involve the creation of functions.
Now whether you can do this using a function based on a “reasonably normal looking combination of operations from math classes up through (say) partial differential equations” – now I have no clue again.
Depending on how you want to interpret Gödel’s incompleteness theorem, maybe? you could probably build two functions that are identical in your intuitioned standard model, but which you can’t prove are identical from your axioms. But in principal, two functions that have the same graph are identical. So if you can prove that they have the same graph then that proof is in and of itself the simplification you’re asking for.
I think Godel has shown how to do that.
F -> true
G -> false
Doesthiscalculationend -> ?
In any sufficiently powerfull mathematical framework we can formulate these 3 functions. The last one is equivalent to one of the other ones, but within the framework we can’t prove which.
Yes, there are, but I can not give you an example. You can show that there are theorms that are true but impossible to prove. You could take a function that is equal to 1 iff such a theorm is true for some value. This function is equal to 1, but you can not show that.
This ultimately comes down to what your words mean:
**Function**:
I think that by “function” you actually mean “formula”, “expression” or “term”. Functions themselves are just the abstract thing, they are the same if and only if they give the same graphs. For example, “sin(x)” is a formula, while the function (or rather its graph) is a sine wave.
A particularly interesting set of formulas are _elementary expressions_: everything you can build from simple numbers, +, -, ·, /, exponentiation, logarithms, and roots.*
*: “simple numbers” is usually implied to mean “complex numbers”, but can be chosen differently.
**Simplification**:
This is the difficult one. If we just go with the functions, not expressions, then there is nothing left to do, your statement is almost vacuously true. However, what you likely mean is to allow _certain_ rules of transforming formulas; of doing algebraic manipulations.
For example, we “know” that sin(x)² and 1-cos(x)² are supposedly the very same thing. So we allow this replacement. Similarly we allow ln( e^x ) = x, sqrt(x²) = |x|, or other rules. The important observation is that it is _us_ who pick the rules here, there is no objective magically given ultimate set of rules.
The _elementary rules_ for example are that – undoes +, / reverses ·, exponentiation and logarithm are inverses, and roots are… roots.
**Some answers**:
It is actually an open unsolved problem if two elementary expressions that always return the same values (“have the same graph”) can be turned into each other using only out elementary rules! So in this case, nobody currently knows!
If we also allow the absolute value function |x|, then things get even worse: we can show that there is definitely no algorithm that can do what we want.
Okay, maybe our stuff was just too complicated with all those logarithms and sines and roots. Lets only use the basic arithmetic of +, -, ·, / and powers. And for those the well-known rules such as a+b=b+a or ( a^^b )^^c = a^^b·c we see in high school. Even then there are things we cannot show, yet are true! This is [Tarski’s High School Algebra Problem](https://en.wikipedia.org/wiki/Tarski%27s_high_school_algebra_problem).
And if one steps further, one even finds issues with the equality of individual numbers…
Yes if the functions are different in dimensions you are not graphing.
For example, a disc and a sphere might both graph a circle in 2D but differ in a third dimension.
No. The mathematical definition of a graph of a function f is the set of ordered pairs (x,f(x)) for all x in the domain. If you have another graph (x,g(x)), and you can show that (x,f(x))=(x,g(x)), then this can be simplified to show that f(x)=g(x).
In logic, your question can be phrased this way: are there functions that are not equivalent intensionally, but equivalent extensionally? In other word, are there functions in which the definitions of the functions are not equivalent, but they happened to refer to the same object anyway?
As it turns out, the answer is yes, there are such thing. However, it’s impossible to give a single explicit example. Why? In order to know that the example work, you need to prove that they are the same function, somehow, by…working with their definition, which means that the 2 functions are equivalent intensionally.
However, even though we could not give a single example, we can give a class of possible example. We can find an infinite list of functions such that one of them is actually equal to 0, but we cannot prove that. In fact, even better, we can have an infinite list of function such that no computer program can determine exactly all the functions that equal 0, even if we let them run as long as they need. A computer program that is allowed to run for as long as it wants would be able to find a proof that a function equal 0 if that’s possible (no matter which axioms you use), so this means that there are no axioms that would give the right answer, no matter what.
How do you obtain such infinite list of function? Look up Richardson’s theorem if you’re curious about the detail. But the key idea is this. We can have an infinite list of function, each function depending on a number, which is the code of a computer program, such that the function is only not zero at some points if the computer program stop running at some points. It’s proven, by Turing, that there are no computer program capable of determining which program will stop running, thus there cannot be a computer program that determine all the functions that are not equivalent to the zero function.
If they generate the same graph, that (generally) means they are equivalent functions. In that case you can interchange them during the process of simplification.
I believe there is a situation where this could be untrue, but it depends if you’re willing to bend your definitions.
Lets say you write a function of x,y, and t. You can mess with the function so that the path along x and y stays the same, but the path along t is shifted or scaled.
You can graph the 2 functions on the x,y plane. They will generate the same graph, but they will not be equivalent functions.
However, this only works if you’re willing to ignore the t variable in your definition of “graphing” the function. Because you could also graph t as a 3rd dimension, and then the graphs wouldn’t be the same anymore.
Another example is a function with a point discontinuity. I forget the form of such a function, but im pretty sure you can have 2 functions with the same graph except one point on one function is undefined. Again, it’s not exactly the same graph, but it’s … very close.
Depends on how you look at it. Definition wise a function *is* its graph, so they’re the same. Just written differently like 1/3 and 2/6 are the same number.
Then again if your function isn’t y=f(x) but (x,y)=f(t) and you consider its graph to be the set of the points (x,y) then there are infinitely many functions with the same graph — differently parameterized.
There are piecewise functions that have names, but you can’t really go from the piecewise definition to the named function without knowing what it is. For example, there’s no algebraic way to go from f(x) = {-x for x<0, x for x>=0} to f(x) = absolute value of x, but they’re the same function by definition.
You can have two functions that produce the same graph visually but have different domains. For example, you can have two functions f(x) and g(x) satisfying both f(x) = x and g(x) = x, but with the former function f defined only over all rational numbers, and the latter function g defined over all real numbers. Since rational numbers are dense (in the set of real numbers), the graphs will look the same at any degree of magnification