Functions

Section 1.2 Functions

The highest function of love is that it makes the loved one a unique and irreplaceable being. — Thomas Eugene “Tom” Robbins, American author, 1932–

The most important term in this course: FUNCTION.

What does the word function mean?

From Oxford Dictionaries:

noun:
1. An activity that is natural to or the purpose of a person or thing:
  
  “bridges perform the function of providing access across water”
2. Mathematics – A relation or expression involving one or more variables:
  
  “the function $(bx + c)$”
3. A thing dependent on another factor or factors:
  
  “class shame is a function of social power”
4. A large or formal social event or ceremony:
  
  “he was obliged to attend party functions”
verb:
1. To work or operate in a proper or particular way:
  
  “her liver is functioning normally”
2. (function as) to fulfil the purpose or task of (a specified thing):
  
  “the museum intends to function as an educational and study centre”

What is the origin of the English word “function”?

Latin fungi = “perform” $\to$ Latin functio $\to$ French fonction $\to$ English function, in the 16th century

How often is the English word “function” used?

Oxford Dictionaries lists the word “function” among the top 1000 frequently used English words.

On May 9, 2021, the website Word frequency data listed the frequency of the word “function” in American English as 1410 (noun) and 3538 (verb).

Why is the word “function” so important in mathematics?

The notion of a function is one of the main tools used in mathematics $\color{red}{\text{to avoid confusion}}\text{.}$

Example 1.2.1. Matchstick puzzle.

Reposition one and remove four matches to spell a thing from which good matches are made. See Figure 1.8.

Figure 1.8. Wood?

Solution

Figure 1.9. Love!

Example 1.2.2. Word puzzle.

Definition of the word pen in Oxford Dictionaries:

An instrument for writing or drawing with ink.
A small enclosure in which sheep, pigs, or other farm animals are kept.
A penitentiary; a prison.
A female swan.

Reminder: Why is the word “function” so important in mathematics?

The notion of a function is one of the main tools used in mathematics $\color{red}{\text{to avoid confusion}}\text{.}$

Example 1.2.3. Canadian or American?

Andrew was born in Vancouver. Is he a Canadian or an American? See Figure 1.10.

Source of the confusion: The statement “Andrew was born in Vancouver” associates Andrew with $\color{red}{\text{more than one}}$ city.

Example 1.2.4. Who is the oldest?

My cousin has three children, Chloe, Justin, and Zack. Chloe is 26 years old and Zack is 28 years old. Who is the oldest among the three siblings? See Figure 1.11.

Figure 1.11. Who is the oldest?

Source of the confusion: $\color{red}{\text{Not each}}$ child is associate with their age.

Function in Mathematics - Definition:

A function is a rule that assigns to each element in a set $A$ exactly one element in a set $B\text{.}$

Example 1.2.5. No confusion.

Andrew was born in Vancouver, BC. Is he a Canadian or an American? See Figure 1.12.

Figure 1.12. A Canadian!
No confusion: The statement “Andrew was born in Vancouver, BC,” associates Andrew with $\color{blue}{\text{only one}}$ city.
My cousin has three children, Chloe, Justin, and Zack. Chloe and Justin are twins and they are 26 years old and Zack is 28 years old. Who is the oldest among the three siblings? See Figure 1.13.

Figure 1.13. Zack is the oldest
No confusion: $\color{blue}{\text{Each}}$ child is associated with their age.

Function in Mathematics - Definition: A function is a rule that assigns to $\color{red}{\text{each element in a set }A}$ $\color{blue}{\text{exactly one element in a set }B}\text{.}$ See Figure 1.14.

Figure 1.14. Function $f:A\to B$

Vocabulary and Notation:

A function $f:A\to B\text{.}$
The set $A=\{-2,-1,0,1,2\}$ is called the domain of the function $f\text{.}$
The set $\{0,1,4\}\subseteq B$ is called the range of the function $f\text{.}$
The rule $f$ is given by:

\begin{equation*} \color{red}{-2}\color{green}{\mapsto}\color{blue}{4},\ \ \color{red}{-1}\color{green}{\mapsto}\color{blue}{1},\ \ \color{red}{0}\color{green}{\mapsto}\color{blue}{0},\ \ \color{red}{1}\color{green}{\mapsto}\color{blue}{1},\ \ \color{red}{2}\color{green}{\mapsto}\color{blue}{4}. \end{equation*}
The rule $f$ is given by:

\begin{equation*} f(-2)=4,\ \ f(-1)=1, \ \ f(0)=0,\ \ f(1)=1,\ \ f(2)=4. \end{equation*}
The rule $f$ is given by:

\begin{equation*} \begin{array}{c|c} x\amp f(x)\\ \hline -2\amp 4\\ -1\amp 1\\ 0\amp 0\\ 1\amp 1\\ 2\amp 4\\ \end{array} \end{equation*}
The rule $f$ is given by:

\begin{equation*} f(x)=x^2, \ \ x\in A=\{ -2,-1,0,1,2\}. \end{equation*}

Think about the function $f$ as a machine. See Figure 1.15.

Figure 1.15. Function $f\text{:}$ A perfect machine

Vocabulary.

Independent variable – a variable that represents input numbers for a function;
Dependent variable – a variable that represents output numbers;
Functional notation – $y=f(x)\text{,}$ which is read “$y$ equals $f$ of $x\text{,}$” means that the output $y$ correspondents to the input $x$ by the rule $f\text{;}$
Range – If $f:A\to B$ then the range of the function $f$ is the set of all elements $y$ in $B$ so that there is an element $x$ in the domain $A$ such that $y=f(x)\text{.}$

For example: If the function $f:\mathbb{R}\to \mathbb{R}$ is defined by $f(x)=x^2$ (where $x$ is a real number), then 9 is in the range of $f$ and $-9$ is not in the range of $f\text{.}$

Example 1.2.6. Functional notation.

Find the domain of each function and calculate $f(2)\text{,}$ $f(-3)\text{,}$ $f(1)\text{,}$ $f(0)\text{,}$ $g(x+1)\text{,}$ and $g(x^2)\text{.}$

$\displaystyle \ds f(x)=\frac{1}{x}$
$\displaystyle \ds g(t)=\sqrt{t}$

Solution

The domain of the function $\ds f(x)=\frac{1}{x}$ is the set $\{x\in \mathbb{R}: x\not=0\}\text{,}$ i.e. the set of all real numbers except zero. (Remember, division by 0 is not defined.)

Also, $\ds f(2)=\frac{1}{2}\text{,}$ $\ds f(-3)=\frac{1}{-3}=-\frac{1}{3}\text{,}$ $\ds f(1)=\frac{1}{1}=1\text{,}$ but $f(0)$ is not defined because the number $0$ does not belong to the domain of the function $f\text{.}$
The domain of the function $\ds g(t)=\sqrt{t}$ is the set of all non-negative real numbers, i.e the interval $[0,\infty)\text{.}$

From $\ds g(t)=\sqrt{t}$ it follows that $\ds g(x+1)=\sqrt{x+1}\text{.}$ Observe that in this case, the number $x+1$ must be in the domain of the function $g\text{,}$ i.e. the number $x$ must be such that $x+1\ge 0\text{.}$ This is the same as $x\ge -1$ or $x\in[-1,\infty)\text{.}$

From $\ds g(t)=\sqrt{t}$ it follows that $\ds g(x^2)=\sqrt{x^2}\text{.}$ Observe that in this case, $x^2\ge 0$ for all real numbers $x\text{.}$ Recall that $\sqrt{x^2}=|x|\text{.}$

Piecewise Defined Functions. The total annual fees in Canadian dollars for students enrolled in a science program at a major university in Montréal, Québec, are as follows:

$\$ 4,214.78\text{,}$ if the student is a resident of Québec;
$\$ 9,226.58\text{,}$ if the student is a Canadian citizen or permanent resident of Canada who is not a resident of Québec;
$\$ 33,325.78\text{,}$ if the student is not a Canadian citizen or permanent resident of Canada.

Example 1.2.7. Three students.

Amir, Biljana, and Candice are enrolled in a math major program at the university mentioned above.

Amir was born and raised in Montréal, Québec. He lives in his parents' house that is just a few blocks away from the university campus.
Biljana represented Bosnia and Herzegovina at the International Math Olympiad earlier this year.
Candice was born and raised in Port Coquitlam, British Columbia. Her plan is to graduate in two years and then to find a job as a data analyst in the Greater Vancouver area.

What is Candice's total annual fee for being enrolled in her program of study? See Figure 1.16.

Figure 1.16. Candice's total annual fee

The same, just a little bit different. Let

$A=\{ x: x$ is a student enrolled in a science program that is a resident of Québec$\}\text{,}$

$B=\{ x: x$ is a student enrolled in a science program that is a not a Canadian citizen or permanent resident of Canada$\}\text{,}$

$C=\{ x: x$ is a student enrolled in a science program that is a Canadian citizen or permanent resident of Canada but is not a resident of Québec$\}\text{.}$

Let $f$ be the rule that assigns to $\color{red}{\text{each student enrolled in a science program}}$ their $\color{blue}{\text{total annual fee in Canadian dollars}}\text{.}$ Then

\begin{equation*} f(x)=\left\{ \begin{array}{rl} 4,214.78,\amp\text{ if } x\in A\\ 33,325.78,\amp\text{ if } x\in B\\ 9,226.58,\amp\text{ if } x\in C\\ \end{array} \right. \end{equation*}

Example 1.2.8. How much?

What is $f(\text{Amir})\text{?}$ What is $f(\text{Dr. J})\text{?}$

Solution

Since Amir is “a student enrolled in a science program that is a resident of Québec,” by the definition of the function $f\text{,}$ it follows that $f(\text{Amir})=4,214.78\text{.}$

Since Dr. J is is not “a student enrolled in a science program,” by the definition of the function $f\text{,}$ the value of the function $f$ at $x = \text{Dr. J}$ is not defined

Two observations and one important new term.

The domain of the function $f$ is the set

\begin{equation*} A\cup B\cup C=\{ x:x\text{ is a student enrolled in a science program}\}. \end{equation*}
The function $f$ is defined by different expressions on different parts of the domain of $f\text{.}$

The function $f$ is a so-called piecewise defined function.

The most famous piecewise defined function:

\begin{equation*} f(x)=\left\{ \begin{array}{rl} x,\amp\text{ if } x\ge 0\\ -x,\amp\text{ if } x\lt 0\\ \end{array} \right. \end{equation*}

Because of its importance and frequent use, the function $f$ is usually written as

\begin{equation*} f(x)=|x|, \end{equation*}

which is read “$f(x)$ equals the absolute value of $x\text{.}$”

Example 1.2.9. Absolute value.

For $f(x)=|x|$ evaluate:

$\displaystyle f(5)=$
$\displaystyle f(0)=$
$\displaystyle f(-5)=$

Solution

Since $-5\lt 0\text{,}$ by definition, $f(-5)=-(-5)=5\text{.}$

By definition, $f(0)=0$

Since $5\gt 0\text{,}$ by definition, $f(5)=5\text{.}$

Two questions that you have to ask.

What is the domain of the function $f(x)=|x|\text{?}$
What is the range of the function $f(x)=|x|\text{?}$

The meaning of things. The function $f(x)=|x|$ is the rule that assigns to $\color{red}{\text{each point }x \text{ on the number line}}$ its $\color{blue}{\text{distance from the origin }0}\text{.}$ See Figure 1.17.

Figure 1.17. Absolute value = Distance from the origin