What is a Function? The Difference between Functions and Relations

What is a Function? The Difference between Functions and Relations exercise

• Describe Herman's problems with the vending machine.

  Hints

  Every function is a relation, but not every relation is a function.

  A function is a special kind of relation where for every input $x$ there is at most one output $y$.

  Solution

  He puts 100 yen in the machine and inserts B3 for the Mammagotcha. But he gets a square melon instead.

  That's because there are several items with the same label, like Mammagotcha and the square watermelon with B3. There is also a label with only one item: S7 and the rocket-toaster.

  With relations, each element in the domain is related to one or more items in the range.

• Explain the difference between functions and relations.

  Hints

  The following relation is a function:

  you $\rightarrow$ number of brothers and sisters you have

  The following relation is not a function:

  you $\rightarrow$ the names of your brothers and sisters

  Every function is a relation, but not the other way round.

  Solution

  Both functions and relations have a set of inputs, called the domain, and a set of outputs, called the range.

  For any relation, for each element of the domain you can have one or more elements in the range.

  For a function, for each element of the domain you can have at most one element in the range.

• Decide which mapping diagrams represent a function.

  Hints

  Keep the definition of a function in mind: for every element of the domain $x$ there exists at most one element in the range $y$ which is assigned to $x$.

  You can imagine the definition of a function as follows: for each element in the domain there is at most one arrow.

  If every $x$ in the domain is assigned to the same $y$, then the function is called a constant function.

  Here is an example of a function:

  Solution

  The kinds of diagrams we are looking at are called mapping diagrams. On the left of each picture we have the domain, the set of inputs, and on the right we have the range, the set of outputs.

  If all inputs $x$ are assigned to at most one output $y$ then the mapping diagram in question is that of a function.

  For any mapping diagram of a function, you see that only one arrow starts at any element of the domain. However many arrows lead to an element in the range doesn't matter.

  Thus, from left to the right, we have a relation, a function, a function, a function, and a relation.

• Identify which statements are describing a function.

  Hints

  For a function, we must have that each input $x$ is assigned at most one output $y$.

  Remember the important facts about the town given above.

  Solution

  For each house in town the address is uniquely assigned. Because of this, we can then view the assignment of addresses to houses as a function! This is how the postman knows where to deliver the mail.

  Specifically, we have the function:

  house $\rightarrow$ address,

  where the address includes the street name, the house number, and the zip code.

  If you leave out any of the three parts of the address, we don't have a function any longer, as the address no longer becomes unique. We know this from the given facts about the town.

  For example:

  • If we leave out the street name, then we know there exists more than one house with the number $30$ in town with the zip code 12345.
  • If we leave out the house number, then we know there exists more than one house on a Beagle Street in the town with the zip code 12345.
  • If we leave out the zip code, then we know there exists more than one house in town on Beagle Street with the house number 30.

• Find three main facts about functions.

  Hints

  Each element of the left set is assigned to one element of the right set.

  The following is an example of a function:

  you $\rightarrow$ your age.

  Keep in mind that not every relation is a function. For example:

  you $\rightarrow$ the names of all your friends.

  Solution

  There are three main facts about functions:

  1. There is a set of all input values $x$, called the domain.
  2. A function $f(x)$ changes an input value $x$ into a unique output value $y$.
  3. The set of all output values $y$ is called the range.

• Determine if the assignment is a function or relation.

  Hints

  A person has a unique hair color but different people could have the same hair color.

  For a function the assignment must be unique.

  Here is an example for the difference between a relation and a function:

  The assignment

  Paul $\rightarrow$ date of his birthday

  is unique and thus a function.

  Turning this assignment around, we get

  date $\rightarrow$ the person who was born on this date

  which isn't unique at all.

  Peter, Paul, and Mary drink just one soda each, but two of them could drink the same kind of soda.

  Solution

  The difference of a relation and a function is the uniqueness of the assignment.

  Hair color

  Each person has a unique hair color, his own, so this assignment is a function. But the other way round you can surely find more than one person with the same color hair. Thus this is a relation.

  Social security card

  Each person has a unique the social security number. So this is a function.

  Email

  Paul has the email addresses paul_1@beagle.us, paul_2@dog.com, and paul_3@quitehungry.net. So three email addresses are assigned to Paul. This is a relation.

  Soda

  Each of the three drinks just one soda, so that's a function. Since two or three of them could order the same soda, this direction is just a relation.