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# A more formal understanding of functions

A more formal understanding of functions. Created by Sal Khan.

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• this video is long!! I'm so confused!! Help, can someone explain what this means? Codomain? Range? •   An example: f(x) = x², x ∈ R. Let's break it down...

f(x) = x²
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f is a function. Think of it like a machine that accepts any number (we use the placeholder x for that number) and produces another number which is the square of the number (x times itself).
x ∈ R
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This means that x is a member of the real numbers. The real number system is a classification of numbers that include whole numbers, negative numbers and and all decimals, so x can be any number. This is called the domain of the function.

(So when you see f(x) = x², x ∈ R you read it as "f of x equals x-squared where x is a member of the reals").

The codomain is also R. This means that our function f spits out numbers from the same classification as the ones it accepts. It so happens that because the numbers are squared that they must be greater than or equal to 0. This is where range comes in... it describes which members of the codomain f will actually output. In our example, the range can be specified using either one of these notations (both of which mean greater than or equal to zero):
0 ≤ f(x) ≤ ∞
[0,∞]
• At , shouldnt the range be {2}, instead of 2. Since the range is a set of numbers, not just a number? • I was just looking at wikipedia and want to be clear on this. Are the R2 and R3 spaces mentioned in this video what are formally called two dimensional and three dimensional Euclidean space respectively? • Danny,

Mostly yes. R2 and R3 could easily represent 2 dimensional and 3 dimensional space, respectively, but they don't HAVE to represent "physical" space. For example, if you have a function (maybe heat transfer rate?) that depends on say, time and temperature, that could be considered "3-dimensional" problem where the three dimensions are time, temperature, and heat transfer rate. You might find it useful to work with the problem in terms of 3x3 matrices and 3x1 vectors, even though the vectors don't really represent vectors in "space" the way you are probably used to thinking about vectors.

In this case, none of the 3 dimensions in the problem represent physical spacial dimensions. So I guess the short answer is R2 and R3 CAN represent two dimensional and 3 dimensional euclidean space, and that is what the often DO represent… they don't HAVE to represent that.
• What's codomain? I know domain and range in kumon, but I don't know what codomain means. Can you tell me a definition, please? • A codomain of a function is any set containing the range of the function - it does not have to equal the range. For example the function y=x² has as a codomain the set of real numbers, which is a set containing the range (y≥0), but is not equal to the range. We can say y=x² maps (sends) the set of real numbers into the set of real numbers, without specifying the range.
• How do I determine the value of the dependent variable when the independent variable is zero. • I am not familiar with the term Sal uses at about . What is a 'tuple'? • Sal used this term also in some of the videos on vectors. The wikipedia entry on Tuple is very helpful. Below are pasted the parts I think are sufficient to get the meaning:

"In mathematics and computer science, a tuple is an ordered list of n elements. In set theory, an (ordered) n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. ...

The term originated as an abstraction of the sequence: single, double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., n‑tuple, ..., where the prefixes are taken from the Latin names of the numerals. The unique 0‑tuple is called the null tuple. A 1‑tuple is called a singleton, a 2‑tuple is called an ordered pair and a 3‑tuple is a triple or triplet. The n can be any nonnegative integer. For example, a complex number can be represented as a 2‑tuple, a quaternion can be represented as a 4‑tuple, an octonion can be represented as an octuple, (many mathematicians write the abbreviation "8‑tuple") and a sedenion can be represented as a 16‑tuple.

Although these uses treat ‑tuple as the suffix, the original suffix was ‑ple as in "triple" (three-fold) or "decuple" (ten‑fold). This originates from a medieval Latin suffix ‑plus (meaning "more") related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded")."
• how to find a gradient of a linear function? • How can you tell from an equation if it's a linear function?   