Functions

Reliable ice-cream machines

3-minute read
Table of Contents

A relation is a relationship between two sets. Relations can be:

  • one-to-one e.g. each student in a class receiving one fruit
  • one-to-many e.g. one drummer playing for many bands (he’s free this Saturday btw)
  • many-to-one e.g. many persons following an artist (they all claim Kes is their husband)
  • many-to-many e.g. citizens reading many newspapers to find a consensus on what is true

The set of all inputs (x values) is referred to as the domain. The set of all outputs (y values) is called the co-domain.

Functions

Functions are relationships between two sets (inputs and outputs) whereby each input has exactly one ($1$) corresponding output. A function can be taught of as a reliable ice-cream machine whereby inputting a certain ingredient will always produce the same result:

Ice-Cream-Machine.png A reliable ice-cream machine (a function)

One input will never produce multiple outputs:

Unreliable-Ice-Cream-Machine.png A unreliable ice-cream machine (just a relation)

The machine above is unreliable because sometimes adding banana gives us banana ice-cream but other times adding banana gives us chocolate ice-cream. It would be better to have a machine that takes banana and produces chocolate ice-cream every time because at least we can depend/rely on it to produce the same output (chocolate ice-cream) every time for the same given input (banana).

Range

The range of a function is the set of the y values produced by the function.

All functions (reliable ice-cream machines) are relations (ice-cream machines) but not all relations (ice-cream machines) are functions (reliable ice-cream machines). Only one-to-one and many-to-one relations are functions.

The vertical line test

In this test, we simply draw a vertical line and if this line cuts the graph of the relation at more than one places, then we conclude that the relation is NOT a function.

Research Questions

Upon completion of this section you will be competent enough to answer the following:

  • What is a relation?
  • What is a function?
  • Are all relations functions?
  • Are all functions relations?
  • What is meant by the domain?
  • What is meant by the range?
  • What do we use the vertical line test for?
  • What is an injective (one-to-one) function?
  • What is the algebraic method of determining if a function is injective?
  • What is the horizontal line test used for?
  • What is a surjective function?
  • What is a bijective function?
  • How do we find the inverse of a function?
  • What is the geometric relationship of the inverse function to the original function?
  • Can we find the inverse of a function which is not bijective?
  • What relationship does the inverse of the inverse of a function have with the function itself?
  • What is meant by the composition of two functions?
  • Can you show that $ff^{-1}(x)=x$ and $f^{-1}f(x)=x$?

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