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Jul 21, 2022 · If $f$ has domain $D$, we can form the set $S = \{f(x) \mid x \in D \}$. This, too, has two names. Some folks call it the "range", and others call it the image. Because of the name-clash with the other use of "range", it's probably a good idea to stick with "codomain" and "image" and leave "range" out of it.
- Why $F^S$ represents set of functions from S to F?
But here's the key fact that I think justifies that...
- Why $F^S$ represents set of functions from S to F?
- Input, Relationship, Output
- Some Examples of Functions
- Names
- The "X" Is Just A Place-Holder!
- Sometimes There Is No Function Name
- Relating
- What Types of Things Do Functions Process?
- A Function Is Special
- The Two Important Things!
- Vertical Line Test
We will see many ways to think about functions, but there are always three main parts: 1. The input 2. The relationship 3. The output
But we are not going to look at specific functions ... ... instead we will look at the general ideaof a function.
First, it is useful to give a function a name. The most common name is "f", but we can have other names like "g" ... or even "marmalade" if we want. But let's use "f": We say "f of x equals x squared" what goes intothe function is put inside parentheses () after the name of the function: So f(x) shows us the function is called "f", and "x" goes in ...
Don't get too concerned about "x", it is just there to show us where the input goes and what happens to it. It could be anything!
Sometimes a function has no name, and we see something like: y = x2 But there is still: 1. an input (x) 2. a relationship (squaring) 3. and an output (y)
At the top we said that a function was likea machine. But a function doesn't really have belts or cogs or any moving parts - and it doesn't actually destroy what we put into it! A function relatesan input to an output. Saying "f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16
So we need something more powerful, and that is where setscome in: Each individual thing in the set (such as "4" or "hat") is called a member, or element. So, a function takes elements of a set, and gives back elements of a set.
But a function has special rules: 1. It must work for everypossible input value 2. And it has only one relationshipfor each input value This can be said in one definition:
When a relationship does not follow those two rules then it is not a function ... it is still a relationship, just not a function.
On a graph, the idea of single valuedmeans that no vertical line ever crosses more than one value. If it crosses more than once it is still a valid curve, but is not a function. Some types of functions have stricter rules, to find out more you can read Injective, Surjective and Bijective
Jan 30, 2024 · Definition: Function. A function is a rule for a relationship between an input, or independent, quantity and an output, or dependent, quantity in which each input value uniquely determines one output value. We say "the output is a function of the input."
A function f from a set X to a set Y is an assignment of one element of Y to each element of X. The set X is called the domain of the function and the set Y is called the codomain of the function. If the element y in Y is assigned to x in X by the function f, one says that f maps x to y, and this is commonly written = ().
Apr 17, 2022 · Define \(F: S \to \mathbb{N}\) by \(F(x) = x^2\) for each \(x \in s\). What is the range of the function \(F\) and what is \(F(S)\)? How do these two sets compare?
Jan 16, 2022 · But here's the key fact that I think justifies that notation: if $F$ and $S$ are finite sets with $f$ and $s$ elements, respectively, then the number of functions from $S$ to $F$ (namely, the number of elements of $F^S$) equals $f^s$.
People also ask
What is a function f on a set S?
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How do you find if two functions are equal?
• We say that two functions f and g are equal if they have (1) the same domain, (2) the same codomain, and if (3) for all x in the domain, f(x) = g(x). • Let f : S → T be a function. The graph of f is the set {(x, f(x)) : x ∈ S}. Notice that the graph is a subset of the Cartesian product S × T . • Let f : S → T be a function.
