5.1 Workshop 1 (Exploration)

5.1.1 Preparatory exercises (30 mins)

Exercise 5.1.

With the usual $x$ - and $y$ -axes in $\mathbb{R}\times\mathbb{R}$ , sketch the curves with the following equations:

(A)

$y=1-x$ ;
(B)

$y=(x-1)^{2}+3$ ;
(C)

$x^{2}+y^{2}=4$ ;
(D)

$y=\sqrt{x}$ ;
(E)

$y=\sin x$ ;
(F)

$x^{2}+y^{2}=-4$ . [Think about what it would mean for a point $(x,y)$ to lie on the curve.]

Be prepared to compare your sketches with your group at the start of the next workshop.

Exercise 5.2.

Consider the following statement:

“The expressions

\frac{1}{1+\sqrt{x}}\mbox{ and }\frac{1-\sqrt{x}}{1-x}

represent the same function.”

Is the statement true or false?

Exercise 5.3.

For any two integers $a$ and $b$ , with $b\neq 0$ , let $f\bigl{(}\frac{a}{b}\bigr{)}=a+b$ .

1.

What is $f\bigl{(}\frac{1}{2}\bigr{)}$ ?
2.

What is $f\bigl{(}\frac{2}{4}\bigr{)}$ ?

5.1.2 B3 Exploration Workshop Supplementary Material

Definition (Function, domain, codomain, graph).

A function is an object consisting of three sets:

(i)

a set $D$ called its domain;
(ii)

a set $C$ called its codomain; and
(iii)

a subset $G$ of $D\times C$ called its graph.

These three sets must have the properties:

(a)

for every $x\in D$ , there exists $y\in C$ such that $(x,y)\in G$ ; and
(b)

if $(x,y)\in G$ and $(x,z)\in G$ then $y=z$ .

Definition (Composition of functions).

Given functions $f:A\to B$ and $g:B\to C$ we can create a new function $g\circ f:A\to C$ defined by $g\circ f:x\mapsto g(f(x))$ .

Definition (Image of a function).

Given a function $f:D\to C$ , the set $\{f(x):x\in D\}$ is called the image of $f$ .

Definition (Image of a subset).

Given a function $f:D\to C$ and a subset $S\subseteq D$ , the set

f(S):=\{f(x):x\in S\}

is called the image of $S$ under $f$ .

Definition (Preimage).

Given a function $f:X\to Y$ , the preimage of $U\subseteq Y$ under $f$ is the subset

f^{-1}(U)=\{x\in X:f(x)\in U\}.

Definition (Periodic function).

Let $D$ be a subset of $\mathbb{R}$ and $f:D\to\mathbb{R}$ be a function. Suppose there is a real number $p>0$ such that $f(x+p)=f(x)$ for all $x\in D$ . Then the function $f$ is said to be $p$ -periodic or periodic with period $p$ .

Definition (Even function).

Let $D$ be a subset of $\mathbb{R}$ . A function $f:D\to\mathbb{R}$ is said to be even if $f(-x)=f(x)$ for all $x\in D$ .

Definition (Odd function).

Let $D$ be a subset of $\mathbb{R}$ . A function $f:D\to\mathbb{R}$ is said to be odd if $f(-x)=-f(x)$ for all $x\in D$ .

Definition (Polynomials and Polynomial Functions).

Suppose $a_{0},a_{1},a_{2},\ldots,a_{n}$ are real numbers, with $a_{n}\neq 0$ . The expression

a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}

is called a polynomial with real coefficients. The numbers $a_{0},a_{1},a_{2},\ldots,a_{n}$ are called coefficients. The integer $n$ is called the degree of the polynomial. The special polynomial $0$ has degree $0$ .

We can use this algebraic expression to define a polynomial function $p:\mathbb{R}\to\mathbb{R}$ given by the formula

p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}.

5.1.3 Workshop tasks

Task 5.4 (5 min).

(Warm Up) Compare your sketches of curves (A)–(F) from the preparatory exercise 5.1 with the rest of your group.

1.

For each equation in this exercise, suppose $(x,y)$ is a pair of real numbers that solves the equation. What values of $x$ are possible? What values of $y$ are possible?
2.

Which of these curves do you think represents a function?

Task 5.5 (10 mins).

Which of the following are functions?

1.

Assume that each $n\in\mathbb{N}$ is written as a product of primes

$n=2^{p_{2}(n)}3^{p_{3}(n)}5^{p_{5}(n)}\cdots,$

where each $p_{k}(n)$ is a non-negative integer, which depends on $n$ . Let $f:\mathbb{N}\to\mathbb{N}_{0}$ be defined as $f(n)=p_{2}(n)$ , where $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$ .

Hint: You may find it helpful to sketch the graph of $f$ for $1\leq n\leq 10$ .
2.

Let $g:\mathbb{Q}\to\mathbb{Z}$ be given by the formula

$g\left(\frac{a}{b}\right)=a+b.$

Hint: Think about your Preparatory Exercise 5.3.
3.

Let $h:\mathbb{Q}\to\mathbb{Z}$ where

$h\left(\frac{a}{b}\right)=\frac{a^{2}+b^{2}}{ab}.$
4.

EXTENSION: Can you think of your own functions $f:\mathbb{Q}\to\mathbb{Z}$ ?

Definition (Composition of functions).

Given functions $f:A\to B$ and $g:B\to C$ we can create a new function $g\circ f:A\to C$ defined by $g\circ f:x\mapsto g(f(x))$ .

Task 5.6 (10 mins).

For a set $X$ , the identity function $id_{X}:X\rightarrow X$ is defined by $id_{X}(x)=x$ for all $x\in X$ .

Let $X=\{1,3\}$ and $Y=\{2,4,5\}$ and let $f:X\rightarrow Y$ and $g:Y\rightarrow X$ be functions.

1.

Complete the definition of functions $f$ and $g$ such that $g\circ f=id_{X}$ .
2.

Is $f\circ g$ also equal to $id_{X}$ ? What if you change your definition of $f$ and $g$ ? Is it possible to define $f$ and $g$ such that $f\circ g=g\circ f=id_{X}$ ?

EXTENSION: If $f:A\to B$ and $g:B\to C$ are functions, is $g\circ f:A\to C$ always a function? What requirements are necessary for $f\circ g:B\rightarrow B$ to be a function?

Definition (Image of a function).

Given a function $f:D\to C$ , the set $\{f(x):x\in D\}$ is called the image of $f$ .

Definition (Image of a subset).

Given a function $f:D\to C$ and a subset $S\subseteq D$ , the set

f(S):=\{f(x):x\in S\}

is called the image of $S$ under $f$ .

Definition (Preimage).

Given a function $f:X\to Y$ , the preimage of $U\subseteq Y$ under $f$ is the subset

f^{-1}(U)=\{x\in X:f(x)\in U\}.

Task 5.7 (15 min).

Let $f:\mathbb{R}\to\mathbb{R}$ with $f(x)=x^{2}$ and let $A=[2,\infty)=B$ .

1.

Find the image of $f$ and the sets $f(A)$ and $f^{-1}(B)$ .
2.

Sketch the graph of $f$ and highlight the sets $f(A)$ and $f^{-1}(B)$ on your sketch.
3.

What is $f^{-1}(f(A))$ ? How does it compare to $A$ ? Is this true for any function $f:X\to Y$ with $A\subseteq X$ ?
4.

What is $f(f^{-1}(B))$ ? How does it compare to $B$ ? Is this true for any function $f:X\to Y$ with $B\subseteq Y$ ?

EXTENSION: Consider a general function $f:X\to Y$ where $A\subseteq X$ and $B\subseteq Y$ . What conditions on $f$ would guarantee that $f^{-1}(f(A))=A$ . What conditions on $f$ would guarantee that $f(f^{-1}(B))=B$ ?

Symmetry of Functions

Definition (Periodic function).

Definition (Even function).

Let $D$ be a subset of $\mathbb{R}$ . A function $f:D\to\mathbb{R}$ is said to be even if $f(-x)=f(x)$ for all $x\in D$ .

Definition (Odd function).

Let $D$ be a subset of $\mathbb{R}$ . A function $f:D\to\mathbb{R}$ is said to be odd if $f(-x)=-f(x)$ for all $x\in D$ .

Task 5.8 (10 mins).

The Dirichlet function $\bm{1}_{\mathbb{Q}}:\mathbb{R}\to\{0,1\}$ is defined by

\bm{1}_{\mathbb{Q}}(x)=\begin{cases}1&\text{if $x\in\mathbb{Q}$}\\ 0&\text{otherwise.}\end{cases}

1.
(a) What is $\bm{1}_{\mathbb{Q}}\bigl{(}\frac{1}{2}\bigr{)}$ ? (b) What is $\bm{1}_{\mathbb{Q}}\bigl{(}\sqrt{7})$ ?
2.

Explain in simple language what the Dirichlet function does.
3.

Is the Dirichlet function odd, even, neither or both?
4.

Is the Dirichlet function periodic? If so, what period does it have?
5.

EXTENSION: Let $a,b\in\mathbb{R}$ . Suppose you have a function $g:\mathbb{R}\to\mathbb{R}$ with period $a$ and $y$ -intercept $b$ . What conditions (if any) are needed on $a,b$ for $g$ to be even? What about for $g$ to be odd?

Definition (Polynomials and Polynomial Functions).

Suppose $a_{0},a_{1},a_{2},\ldots,a_{n}$ are real numbers, with $a_{n}\neq 0$ . The expression

a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}

We can use this algebraic expression to define a polynomial function $p:\mathbb{R}\to\mathbb{R}$ given by the formula

p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}.

Task 5.9 (10 min).

Consider a general cubic polynomial with $a\neq 0$ and $b,c,d\in\mathbb{R}$ :

p(x)=ax^{3}+bx^{2}+cx+d.

1.

Find the expanded form of $p(x+s)$ where $s\in\mathbb{R}$ is some constant. How does the graph of $p(x+s)$ relate to the graph of $p(s)$ ?
2.

Note that $p(x)=p((x-s)+s)$ . Hence express $p(x)$ in terms of powers of $(x-s)$ . How do the coefficients of the $(x-s)$ terms relate to the derivatives of $p(x)$ ?
3.

What happens if we set $s=-\frac{b}{3a}$ ( $s$ is said to be the point of inflexion of the cubic)? Use this to form a conjecture about the symmetry of cubic polynomials.
4.

EXTENSION: Use your workings to construct a formal proof for your conjecture.