A more formal understanding of functions | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

Functions are a way to relate two sets by mapping each element of one set to another element in the other set. This video explains the concept of functions and their notation, as well as introduces the terms domain, codomain, and range.

Transcript

I think you've been exposed to the idea of a function at some point in your mathematical career. But what I want to do in this video is explain it a little bit more formerly than you might be used to, and then relate it to some of the concepts of vectors and linear algebra that we've seen so far. A function really is just a relation between the mem... Read More

Key Insights

• 😫 Functions establish a relationship between two sets by mapping each element of one set to another element in the other set.
• ❣️ Functions can be represented using notation such as f(x) = y, where x is an input from the domain and y is the corresponding output in the codomain.
• 😫 The domain of a function determines which values are valid inputs, while the codomain is the set to which the function can map.
• 🧡 The range is the subset of the codomain that is actually reached by the function, containing all the possible outputs or values that the function can produce.
• 👾 Scalar valued functions map to one-dimensional spaces, while vector valued functions map to spaces with more than one dimension.

Questions & Answers

Q: What is a function in mathematics?

A function is a mathematical relation between two sets, where each member of the first set is associated with a member of the second set.

Q: How is a function represented in notation?

A function can be represented as f(x) = y, where x is an element from the domain set and y is the corresponding element in the codomain set.

Q: What is the domain of a function?

The domain of a function is the set of all valid inputs or arguments for the function. It is part of the function definition and determines which values can be used as input.

Q: How does the codomain relate to the range?

The codomain is the set to which the function can map, while the range is the subset of the codomain that the function actually reaches. The range is the set of all possible outputs or values that the function produces.

Q: What is the difference between scalar valued and vector valued functions?

Scalar valued functions map to one-dimensional spaces (typically real numbers) and are the most common type of function. Vector valued functions map to spaces with more than one dimension (e.g., R2, R3) and are used in vector calculus.

Summary & Key Takeaways

• Functions are a relation between two sets, where each element of one set is associated with an element of the other set.

• Functions can be represented using a notation that shows the mapping between the sets.

• The terms domain, codomain, and range are used to describe the set from which the inputs are taken, the set to which the outputs are mapped, and the subset of the codomain that is actually reached by the function, respectively.