What really is a function?

We look at the precise definition of a function.

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[MichaelPennMath]

Two things:
1. New studio space is still under construction.
2. not every transition can be super smooth and cinematic...

[taufiqutomo]

One underrated thing he did: The slides are friendly to screens of even the smallest of phones.

Edit: even in 144p

[caldersheagren]

7:40 Isn't x restricted to be positive? That would mean (1, -1) \not\in h ...

[goodplacetostop]

3:43 Now this is production value
9:43 Good Place To Stop

[tomkerruish]

7:16. I'm confused. The way h is written, it can't contain (1, -1) since x can't be negative.

[lucakoch]

Awesome video! Also love that you‘re experimenting with different locations and camera shots, looks great! Love your content. Greetings from Germany ^^

[EpicMathTime]

I don't think your leading definition is quite correct as displayed. I think you should say "such that (a, b) \in f" at the end. Otherwise, your statement about a and b do not actually involve the function.

[akashpoudel]

hey Michael Penn, just letting you know you are amazing ❤,
Your videos really help me understand the concepts so I'd like to encourage you to make more videos on the topics of mathematics. Your videos have helped me when I didn't understand the concepts in calculus and real analysis.
Thank you so much!

[Chess-kslk]

I would like to buy a full academic course from you. Have u considered making such courses?

[oida]

Isn't h={(x^2, x) | x in [0, infty)} still a function since (1, -1) would not be in there as -1 not in [0, infty)?

[roberttelarket]

A definition for a function will vary with certain mathematicians even though the core meaning is the same. All you need are two sets(non-empty preferred). If W is to be a function all you need is again two non-empty subsets one called the pre-image and the other the image such that if p is in the pre-image it is "mapped" to only one element from the image. Otherwise it is called a multi-valued function.

[samdean]

Your definition should say (a, b) is in f, not in AxB

[gaborendredi]

I am really confused or I missed something in the last but one example. I see there a condition that x >= 0, therefore for me h(1)=-1 is not a counterexample. What did I miss?

[AM-hjbo]

Love the outdoor blackboard. Do you use Hagaromo chalk?

[shivam]

Question: at 7:35, you took the set (1, -1) and (1, 1) however, the first one does not satisfy the condition (x^2, x), actually they would be (-1, 1) and(1, 1) respectively, which is a set since no element in the domain is assigned to 2 different elements in the co domain

[natepolidoro]

Hello I think you're videos are great and very interesting.

[MaxG]

At the bottom of the first slide, I think you meant “is an element of f” not “is an element of a cross b”.

[CodingDragon]

What always doesn't quite work for me with this definition of a function, is that a function f:A -> B would be identical to a function g:A -> C, x |-> f(x), where B is a subset of C. I thought that functions with different codomains should be considered different.

[nahuelcaruso]

I don't know if the definition is ok, I think it would end with (a, b) in f

[iabervon]

The first one is a function, but not one whose domain is the integers like it was presumably supposed to be. We can guess what the domain ought to be from the form of the definition, but there's nothing in the set of pairs to let us know that 1 is supposed to have a value.