Video 7: Introduction to Functions by MIT OpenCourseWare

Summary by www.lecturesummary.com: Video 7: Introduction to Functions by MIT OpenCourseWare OpenCourseWare 

• A Brief Overview of Functions (around 0:00)

  • Functions are a key and essential idea for understanding many mathematical concepts and for engineering entrance exams.
  • Many physical, chemical, and real-world phenomena are described by them.
  • Understanding functions is crucial for later subjects like limits, derivatives, and continuity.
  • Functions are very helpful for future topics and have a variety of definitions.

  Intuitive Definition of Functions (around 0:45)

  A special relation between two variables is described by a function.

  • Determining the relationship between two variables of interest is an intuitive concept.
  • This is significant because systems frequently have inputs and outputs, and functions aid in determining the actions of the system.

  Methods of Describing Functions (around 1:15)

  • A graph is usually used to describe functions.
  • It is easy to describe functions using a rule.
  • An equation is frequently used to describe functions, especially as you gain more knowledge.

  Example 1: Sonu's Day (around 2:10)

  The life of a student attending class and preparing for an exam is depicted in this example.

  • The time of day (Variable 1) and Sonu's distance from home (Variable 2) are the variables of interest.
  • A plot or graph with time on the x-axis and distance from home on the y-axis can illustrate the relationship between these variables.
  • The relationship between the two variables over the course of the day is intuitively described by this plot.

  Example 2: Constant Velocity Object (around 3:50)

  This example is from the study of an object moving at a constant speed in physics.

  • Once more, time (Variable 1) and distance from the starting point (Variable 2) are the variables.
  • An equation, such as D = 1 * t, where D is distance and t is time, can describe the relationship if the object moves at a constant speed.

  Interchangeability of Descriptions (around 4:50)

  • Functions can be described by an equation, a rule, or a graph, and these descriptions are interchangeable.
  • An equation can be represented as a graph, and a rule can be expressed as an equation.

  Functions as Input-Output Relation (around 5:20)

  • Functions can also be thought of as input-output relations.
  • You can think of it as a system that creates an output by passing an input through a function.
  • The Sonu example used the time of day as the input and the distance from home as the output.
  • In the object example, the input was the time in hours, and the output was the distance from the starting point.
  • Functions can be used to describe anything that transforms an input into an output.

  Usefulness and Additional Examples (around 6:10)

  • Functions are very helpful and can be used to determine a system's properties.
  • Functions are helpful in characterizing practically all physical and chemical phenomena.
  • The time after a ball was hit and its speed are two examples from everyday life or sports.
  • Another example would be taking a train, where the number of passengers could be the output variable (Variable 2) and the distance from the starting point could be the input variable (Variable 1).

  Plotting Convention (around 7:55)

  • Variables 1 and 2 are usually positioned on the x-axis and y-axis, respectively, in graphs.
  • Variable 1 is frequently regarded as the input and Variable 2 as the output, which is one explanation for this convention.

  Mathematically Precise Definition Transition (approximately 8:25)

  • Functions have so far been defined intuitively in the lecture. Examining functions from a mathematically precise perspective is the next step.

    Key Considerations

    • Returning to the assertion that functions are a unique relationship between two variables will be necessary.

    Mathematical Concepts for Precise Definition

    Knowing a few mathematical terms is necessary to comprehend the exact definition:

    • Variables and relations are essential concepts.
    • These pertain to ideas like sets that you may already be familiar with.
    • A set is a group of things. Assume we have sets A and B.
    • These sets are the source of variables.
    • The Cartesian product of two sets (A x B) contains every possible ordered pair of elements from A and B. The number of elements in A multiplied by the number of elements in B gives the number of elements in A x B.
    • In mathematics, a relation is a subset of the Cartesian product of two sets.

    The next lecture will describe how to determine what makes functions "special" by combining sets, Cartesian products, and particularly relations.