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Beginning Activity Beginning Activity 2: Some Other Types of Functions
A4US

The domain and codomain of each of the functions in Beginning Activity 1 are the set \(\R\) of all real numbers, or some subset of \(\R\text{.}\) In most of these cases, the way in which the function associates elements of the domain with elements of the codomain is by a rule determined by some mathematical expression. For example, when we say that \(f\) is the function such that

\begin{equation*} f( x ) = \dfrac{x}{{x - 1}}\text{,} \end{equation*}

then the algebraic rule that determines the output of the function \(f\) when the input is \(x\) is \(\dfrac{x}{{x - 1}}\text{.}\) In this case, we would say that the domain of \(f\) is the set of all real numbers not equal to 1 since division by zero is not defined.

However, the concept of a function is much more general than this. The domain and codomain of a function can be any set, and the way in which a function associates elements of the domain with elements of the codomain can have many different forms. The input-output rule for a function can be a formula, a graph, a table, a random process, or a verbal description. We will explore two different examples in this activity.

1.

Let \(b\) be the function that assigns to each person his or her birthday (month and day). The domain of the function \(b\) is the set of all people and the codomain of \(b\) is the set of all days in a leap year (i.e., January 1 through December 31, including February 29).

(a)

Explain why \(b\) really is a function. We will call this the birthday function.

(b)

In 1995, Andrew Wiles became famous for publishing a proof of Fermat's Last Theorem. (See A. D. Aczel, Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem, Dell Publishing, New York, 1996.) Andrew Wiles's birthday is April 11, 1953. Translate this fact into functional notation using the ā€œbirthday functionā€ \(b\text{.}\) That is, fill in the spaces for the following question marks:

\begin{equation*} b( {\,?\,} ) = \,?\text{.} \end{equation*}
(c)

Is the following statement true or false? Explain.

For each day \(D\) of the year, there exists a person \(x\) such that \(b( x ) = D\text{.}\)
(d)

Is the following statement true or false? Explain.

For any people \(x\) and \(y\text{,}\) if \(x\) and \(y\) are different people, then \(b( x ) \ne b( y )\text{.}\)

2.

Let \(s\) be the function that associates with each natural number the sum of its distinct natural number divisors. This is called the sum of the divisors function. For example, the natural number divisors of 6 are 1, 2, 3, and 6, and so

\begin{align*} s( 6 ) \amp = 1 + 2 + 3 + 6\\ \amp = 12\text{.} \end{align*}
(a)

Calculate \(s( k )\) for each natural number \(k\) from 1 through 15.

(b)

Does there exist a natural number \(n\) such that \(s( n ) = 5\text{?}\) Justify your conclusion.

(c)

Is it possible to find two different natural numbers \(m\) and \(n\) such that \(s( m ) = s( n )\text{?}\) Explain.

(d)

Use your responses in TaskĀ 2.b TaskĀ 2.c to determine the truth value of each of the following statements.

(i)

For each \(m \in \mathbb{N}\text{,}\) there exists a natural number \(n\) such that \(s( n ) = m\text{.}\)

(ii)

For all \(m, n \in \mathbb{N}\text{,}\) if \(m \ne n\text{,}\) then \(s( m ) \ne s( n )\text{.}\)