02-functions.Rmd

# Functions and Notation

__Topics__
 Dimensionality;
 Interval Notation for ${\bf R}^1$;
 Neighborhoods: Intervals, Disks, and Balls; Introduction to Functions;
 Domain and Range;
 Some General Types of Functions;
 $\log$, $\ln$, and $\exp$;
 Other Useful Functions;
 Graphing Functions;
 Solving for Variables;
 Finding Roots;
 Limit of a Function;
 Continuity; Sets, Sets, and More Sets.
 
Much of the material and examples for this lecture are taken from Simon \& Blume (1994) \emph{Mathematics for Economists}, Boyce \& Diprima (1988) \emph{Calculus}, and Protter \& Morrey (1991) \emph{A First Course in Real Analysis}


## Dimensionality


${\bf R}^1$ is the set of all real numbers extending from $-\infty$ to $+\infty$ --- i.e., the real number line. ${\bf R}^n$ is an $n$-dimensional space (often referred to as Euclidean space), where each of the $n$ axes extends from $-\infty$ to $+\infty$.

* ${\bf R}^1$ is a one dimensional line.
* ${\bf R}^2$ is a two dimensional plane.
* ${\bf R}^3$ is a three dimensional space.
* ${\bf R}^4$ could be 3-D plus time (or temperature, etc).

Points in ${\bf R}^n$ are ordered $n$-tuples, where each element of the $n$-tuple represents the coordinate along that dimension.

* ${\bf R}^1$: (3)
* ${\bf R}^2$: (-15, 5)
* ${\bf R}^3$: (86, 4, 0)
	


## Interval Notation for ${\bf R}^1$

Open interval: $$(a,b)\equiv \{ x\in{\bf R}^1: a<x<b\}$$

$x$ is a one-dimensional element in which x is greater than a and less than b

Closed interval: $$[a,b]\equiv \{ x\in{\bf R}^1: a\le x \le b\}$$

$x$ is a one-dimensional element in which x is greater or equal to than a and less than  or equal to b

Half open, half closed: $$(a,b]\equiv \{ x\in{\bf R}^1: a<x\le b\}$$

$x$ is a one-dimensional element in which x is greater than a and less than or equal to b


## Neighborhoods: Intervals, Disks, and Balls

In many areas of math, we need a formal construct for what it means to be "near" a point $\bf c$ in ${\bf R}^n$.  This is generally called the __neighborhood__ of $\bf c$. It's represented by an open interval, disk, or ball, depending on whether ${\bf R}^n$ is of one, two, or more dimensions, respectively.

Given the point $c$, these are defined as

* $\epsilon$-interval in ${\bf R}^1$: $\{x : |x-c|<\epsilon \}$. $x$ is in the neighborhood of {\bf c} if it is in the open interval $(c-\epsilon,c+\epsilon)$.
* $\epsilon$-disk in ${\bf R}^2$: $\{x : || x-c ||<\epsilon\}$. $x$ is in the neighborhood of {\bf c} if it is inside the circle or disc with center $\bf c$ and radius $\epsilon$.
* $\epsilon$-ball in ${\bf R}^n$: $\{x : || x-c ||<\epsilon\}$. $x$ is in the neighborhood of {\bf c} if it is inside the sphere or ball with center $\bf c$ and radius $\epsilon$.



## Introduction to Functions

A __function__ (in ${\bf R}^1$) is a mapping, or transformation, that relates members of one set to members of another set. For instance, if you have two sets: set $A$ and set $B$, a function from $A$ to $B$ maps every value $a$ in set $A$ such that $f(a) \in  B$. Functions can be "many-to-one", where many values or combinations of values from set $A$ produce a single output in set $B$, or they can be "one-to-one", where each value in set $A$ corresponds to a single value in set $B$.

Examples: Mapping notation 

* Function of one variable: $f:{\bf R}^1\to{\bf R}^1$\\
    $f(x)=x+1$. For each $x$ in ${\bf R}^1$, $f(x)$ assigns the number $x+1$.
* Function of two variables: $f: {\bf R}^2\to{\bf R}^1$. $f(x,y)=x^2+y^2$. For each ordered pair $(x,y)$ in ${\bf R}^2$, $f(x,y)$ assigns the number $x^2+y^2$.

We often use variable $x$ as input and another $y$ as output, e.g. $y=x+1$


## Domain and Range/Image

Some functions are defined only on proper subsets of ${\bf R}^n$.

* __Domain__: the set of numbers in $X$ at which $f(x)$ is defined.
* __Range__: elements of $Y$ assigned by $f(x)$ to elements of $X$, or $$f(X)=\{ y : y=f(x), x\in X\}$$
	Most often used when talking about a function $f:{\bf R}^1\to{\bf R}^1$.
* __Image__: same as range, but more often used when talking about a function $f:{\bf R}^n\to{\bf R}^1$.
	
<!-- \begin{samepage}	 -->
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<!-- 		\item \parbox[c]{3.75in}{$f(x)=\frac{3}{1+x^2}\quad$ \\[6pt] -->
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<!-- 				1-x,  &      & -2\le x\le -1 -->
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<!-- 			%Domain $X=$	$[-2,-1]\cup\{0\}\cup[1,2] -->
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<!-- 		\item \parbox[c]{3.75in}{$f(x)=1/x$\\[6pt] -->
<!-- 			%Domain $X=(-\infty, 0)\cup (0,\infty)$\\ -->
<!-- 			%Range $f(X)=(-\infty, 0)\cup (0,\infty)$ -->
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<!-- 		\item \parbox[c]{3.75in}{$f(x,y)=x^2+y^2$\\[6pt] -->
<!-- 			%Domain $X,Y={\bf R}^2$\\ -->
<!-- 			%Image $f(X,Y)={\bf R}^1_+$ -->
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## Some General Types of Functions
__Monomials__:  $f(x)=a x^k$\\
		$a$ is the coefficient.  $k$ is the degree.\\
		Examples: $y=x^2$, $y=-\frac{1}{2}x^3$
	
__Polynomials__: sum of monomials. Examples: $y=-\frac{1}{2}x^3+x^2$, $y=3x+5$

The degree of a polynomial is the highest degree of its monomial terms.  Also, it's often a good idea to write polynomials with terms in decreasing degree.
	
__Rational Functions__: ratio of two polynomials.

Examples: $y=\frac{x}{2}$, $y=\frac{x^2+1}{x^2-2x+1}$

__Exponential Functions__: Example: $y=2^x$

__Trigonometric Functions__: Examples: $y=\cos(x)$, $y=3\sin(4x)$

\begin{comment}
	\parbox[c]{4.75in}{{\bf Linear}: polynomial of degree 1.\\
		Example: $y=m x + b$, where $m$ is the slope and $b$ is the $y$-intercept.}\epsfxsize=1in \parbox{1in}{\, {\includegraphics[width=.9in, angle = 270]{linear.eps}}}
	
	\item \parbox[c]{4.75in}{{\bf Nonlinear}: anything that isn't constant or polynomial of degree 1.\\
		Examples:  $y=x^2+2x+1$, $y=\sin(x)$, $y=\ln(x)$, $y=e^x$}
		\parbox{1in}{\,  {\includegraphics[width=.9in, angle = 270]{nonlin.eps}}}
\end{comment}

## $\log$, $\ln$, and $\exp$

__Relationship of logarithmic and exponential functions__: 
		$$y=\log_a(x) \iff a^y=x$$

The log function can be thought of as an inverse for exponential functions. $a$ is referred to as the "base" of the logarithm.
	
__Common Bases__: The two most common logarithms are base 10 and base $e$.
	
1. Base 10: $\quad y=\log_{10}(x) \iff 10^y=x$. The base 10 logarithm is often simply written as "$\log(x)$" with no base denoted.
2. Base $e$: $\quad y=\log_e(x) \iff e^y=x$. The base $e$ logarithm is referred to as the "natural" logarithm and is written as ``$\ln(x)$". 

\begin{comment}
			{\includegraphics[width=1in, angle = 270]{ln.eps}} \,  {\includegraphics[width=1in, angle = 270]{exp.eps}}
			\end{comment}
			
__Properties of exponential functions:__
	
* $a^x a^y = a^{x+y}$
* $a^{-x} = 1/a^x$
* $a^x/a^y = a^{x-y}$
* $(a^x)^y = a^{x y}$
* $a^0 = 1$
	
	
__Properties of logarithmic functions__ (any base):

Generally, when statisticians or social scientists write $\log(x)$ they mean $\log_e(x)$. In other words: $\log_e(x) \equiv \ln(x) \equiv \log(x)$
			
$$\log_a(a^x)=x$$ and 
$$a^{\log_a(x)}=x$$
	
* $\log(x y)=\log(x)+\log(y)$
* $\log(x^y)=y\log(x)$
* $\log(1/x)=\log(x^{-1})=-\log(x)$
* $\log(x/y)=\log(x\cdot y^{-1})=\log(x)+\log(y^{-1})=\log(x)-\log(y)$
* $\log(1)=\log(e^0)=0$
		
		
__Change of Base Formula__: Use the change of base formula to switch bases as necessary: 
$$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$$

Example: $$\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$$

\begin{itemize}
\item $\log_{10}(\sqrt{10})=\log_{10}(10^{1/2}) = $
\item $\log_{10}(1)=\log_{10}(10^{0}) = $
\item $\log_{10}(10)=\log_{10}(10^{1}) = $
\item $\log_{10}(100)=\log_{10}(10^{2}) = $
\item $\ln(1)=\ln(e^{0}) = $
\item $\ln(e)=\ln(e^{1}) = $
\end{itemize}


## Other Useful Functions
__Factorials!:__

$$x! = x\cdot (x-1) \cdot (x-2) \cdots (1)$$

__Modulo:__ Tells you the remainder when you divide one number by another. Can be extremely useful for programming: \texttt{x mod y} or \texttt{x \% y}

* $17 \mod 3 = 2$
* $100 \ \% \ 30 = 10$

__Summation:__

$$\sum\limits_{i=1}^n x_i = x_1+x_2+x_3+\cdots+x_n$$

\begin{itemize}
\item $\sum\limits_{i=1}^n c x_i = c \sum\limits_{i=1}^n x_i $
\item $\sum\limits_{i=1}^n (x_i + y_i) =  \sum\limits_{i=1}^n x_i + \sum\limits_{i=1}^n y_i $
\item $\sum\limits_{i=1}^n c = n c $
\end{itemize}

__Product:__

$$\prod\limits_{i=1}^n x_i = x_1 x_2 x_3 \cdots x_n$$

Properties:

\begin{itemize}
\item $\prod\limits_{i=1}^n c x_i = c^n \prod\limits_{i=1}^n x_i $
\item $\prod\limits_{i=1}^n (x_i + y_i) =$ a total mess
\item $\prod\limits_{i=1}^n c = c^n $
\end{itemize}


You can use logs to go between sum and product notation. This will be particularly important when you're learning maximum likelihood estimation.

\begin{eqnarray*}
			\log \bigg(\prod\limits_{i=1}^n x_i \bigg) &=& \log(x_1 \cdot x_2 \cdot x_3 \cdots \cdot x_n)\\
			&=& \log(x_1) + \log(x_2) + \log(x_3) + \cdots + \log(x_n)\\
			&=& \sum\limits_{i=1}^n \log (x_i)
\end{eqnarray*}
		
Therefore, you can see that the log of a product is equal to the sum of the logs. We can write this more generally by adding in a constant, $c$:
		
\begin{eqnarray*}
			\log \bigg(\prod\limits_{i=1}^n c x_i\bigg) &=& \log(cx_1 \cdot cx_2 \cdots cx_n)\\
			&=& \log(c^n \cdot x_1 \cdot x_2 \cdots x_n)\\
			&=& \log(c^n) + \log(x_1) + \log(x_2) + \cdots + \log(x_n)\\\\
			&=& n \log(c) +  \sum\limits_{i=1}^n \log (x_i)\\
\end{eqnarray*}	


## Graphing Functions
What can a graph tell you about a function?

* Is the function increasing or decreasing?  Over what part of the domain?
* How ``fast" does it increase or decrease?
* Are there global or local maxima and minima?  Where?
* Are there inflection points?
* Is the function continuous?
* Is the function differentiable?
* Does the function tend to some limit?
* Other questions related to the substance of the problem at hand.


## Solving for Variables and Finding Inverses
	
Sometimes we're given a function $y=f(x)$ and we want to find how $x$ varies as a function of $y$. If $f$ is a one-to-one mapping, then it has an inverse.

Use algebra to move $x$ to the left hand side (LHS) of the equation and so that the right hand side (RHS) is only a function of $y$.\\

Examples:  (we want to solve for $x$)

$$y=3x+2 \quad\Longrightarrow\quad -3x=2-y \quad\Longrightarrow\quad 3x=y-2 \quad\Longrightarrow\quad x=\frac{1}{3}(y-2)$$
$$y=3x-4z+2 \quad \Longrightarrow\quad y+4z-2=3x \quad\Longrightarrow\quad x=\frac{1}{3}(y+4z-2)$$

\begin{align*}
y=e^x+4 &\Longrightarrow\quad y-4=e^x\\
&\Longrightarrow \ln(y-4) = \ln(e^x)\\
&\Longrightarrow\quad x=\ln(y-4)
\end{align*}


Sometimes (often?) the inverse does not exist. For example, we're given the function $y=x^2$ (a parabola).  Solving for $x$, we get $x=\pm\sqrt{y}$. For each value of $y$, there are two values of $x$. 


## Finding the Roots or Zeroes of a Function

Solving for variables is especially important when we want to find the __roots__ of an equation: those values of variables that cause an equation to equal zero. Especially important in finding equilibria and in doing maximum likelihood estimation.

Procedure: Given $y=f(x)$, set $f(x)=0$.  Solve for $x$.

Multiple Roots:
		$$f(x)=x^2 - 9 \quad\Longrightarrow\quad 0=x^2 - 9 \quad\Longrightarrow\quad 9=x^2 \quad\Longrightarrow\quad \pm \sqrt{9}=\sqrt{x^2} \quad\Longrightarrow\quad \pm 3=x$$

__Quadratic Formula:__ For quadratic equations $ax^2+bx+c=0$, use the quadratic formula: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
	
Examples:
\begin{enumerate}
		\item $f(x)=3x+2$ \\
		% $\quad\Longrightarrow\quad 3x+2=0 
		% \quad\Longrightarrow\quad x=-\frac{2}{3}$\\
		\item $f(x)=e^{-x}-10$ \\
		%$\quad\Longrightarrow\quad e^{-x}-10=0 
		%\quad\Longrightarrow\quad e^{-x}=10 \quad\Longrightarrow\quad 
		%x=-\ln(10)$\\
		\item $f(x)=x^2+3x-4=0$  \\
		%$\quad\Longrightarrow\quad x=\{1,-4\}$\\
\end{enumerate}

## The Limit of a Function
	
We're often interested in determining if a function $f$ approaches some number $L$ as its independent variable $x$ moves to some number $c$ (usually 0 or $\pm\infty$).  If it does, we say that the limit of $f(x)$, as $x$ approaches $c$, is $L$: $\lim\limits_{x \to c} f(x)=L$.
	
For a limit $L$ to exist, the function $f(x)$ must approach $L$ from both the left and right.
	
__Limit of a function__.  Let $f(x)$ be defined at each point in some open interval containing the point $c$.  Then $L$ equals $\lim\limits_{x \to c} f(x)$ if for any (small positive) number $\epsilon$, there exists a corresponding number $\delta>0$ such that if $0<|x-c|<\delta$, then $|f(x)-L|<\epsilon$.

Note: f(x) does not necessarily have to be defined at $c$ for $\lim\limits_{x \to c}$ to exist.

__Uniqueness__: $\lim\limits_{x \to c} f(x)=L$ and $\lim\limits_{x \to c} f(x)=M \Longrightarrow L=M$

Properties: Let $f$ and $g$ be functions with $\lim\limits_{x \to c} f(x)=A$ and $\lim\limits_{x \to c} g(x)=B$.


* $\lim\limits_{x \to c}[f(x)+g(x)]=\lim\limits_{x \to c} f(x)+ \lim\limits_{x \to c} g(x)$
* $\lim\limits_{x \to c} \alpha f(x) = \alpha \lim\limits_{x \to c} f(x)$
* $\lim\limits_{x \to c} f(x) g(x) = [\lim\limits_{x \to c} f(x)][\lim\limits_{x \to c} g(x)]$
* $\lim\limits_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to c} f(x)}{\lim\limits_{x \to c} g(x)}$, provided $\lim\limits_{x \to c} g(x)\ne 0$.

Note: In a couple days we'll talk about L'H\^opital's Rule, which uses simple calculus to help find the limits of functions like this.

Examples:

\begin{align*}
& \lim_{x \to c} k = \\
& \lim_{x \to c} x = \\
& \lim_{x\to 2} (2x-3) =\\
& \lim_{x \to c} x^n = 
\end{align*}

\begin{comment}
%= 2\lim\limits_{x\to 2} x- 3\lim\limits_{x\to 2} 1 = 2\times 2 - 3\times 1 = 1 
%[\lim\limits_{x \to c} x]\cdots[\lim\limits_{x \to c} x] = c\cdots c =c^n 
\end{comment}


\begin{comment}
		\item \parbox[t]{3.75in}{$\lim\limits_{x\to 0} |x| =$} \parbox[t]{1in}{\,{\includegraphics[width=1in, angle = 270]{abs.eps}}} % = 0
		\item \parbox[t]{3.75in}{$\lim\limits_{x\to 0} \left(1+\frac{1}{x^2}\right)=$} \parbox[t]{1in}{\,  {\includegraphics[width=1in, angle = 270]{1p1ovrx2.eps}}} %\infty
\end{comment}
		


Types of limits:

1. Right-hand limit: The value approached by $f(x)$ when you move from right to left.
\begin{comment}
		\parbox[t]{2in}{Example:  $\lim\limits_{x\to 0^+} \sqrt{x} = 0$}\parbox[t]{1in}{\,  {\includegraphics[width=1in, angle = 270]{sqrt.eps}}}
		\end{comment}
2. Left-hand limit: The value approached by $f(x)$ when you move from left to right.
3. Infinity: The value approached by $f(x)$ as x grows infinitely large. Sometimes this may be a number; sometimes it might be $\infty$ or $-\infty$.
4. Negative infinity: The value approached by $f(x)$ as x grows infinitely negative. Sometimes this may be a number; sometimes it might be $\infty$ or $-\infty$.

\begin{comment}
\parbox[t]{2in}{Example:  $\lim\limits_{x\to \infty} 1/x = \lim\limits_{x\to -\infty} 1/x= 0$}\parbox[t]{1in}{\,  {\includegraphics[width=1in, angle = 270]{1ovrx.eps}}}\\ 
\end{comment}


## Continuity

__Continuity__: Suppose that the domain of the function $f$ includes an open interval containing the point $c$.  Then $f$ is continuous at $c$ if $\lim\limits_{x \to c} f(x)$ exists and if $\lim\limits_{x \to c} f(x)=f(c)$. Further, $f$ is continuous on an open interval $(a,b)$ if it is continuous at each point in the interval.


Examples: Continuous functions.

\begin{comment}
	\parbox[t]{1.5in}{\hfill$f(x)=\sqrt{x}\quad$}\parbox[t]{1in}{\,  {\includegraphics[width=1in, angle = 270]{sqrt.eps}}}
	\parbox[t]{1.5in}{\hfill$f(x)=e^x\quad$}\parbox[t]{1in}{\,  {\includegraphics[width=1in, angle = 270]{exp.eps}}}
	\item[] Examples: Discontinuous functions.\\
	\parbox[t]{1.5in}{\hfill$f(x)=\mbox{floor}(x)\quad$}\parbox[t]{1in}{\,  {\includegraphics[width=1in, angle = 270]{floor.eps}}}
	\parbox[t]{1.5in}{\hfill$f(x)=1+\frac{1}{x^2}\quad$}\parbox[t]{1in}{\,  {\includegraphics[width=1in, angle = 270]{1p1ovrx2.eps}}}
\end{comment}

Properties:

1. If $f$ and $g$ are continuous at point $c$, then $f+g$, $f-g$, $f \cdot g$, $|f|$, and $\alpha f$ are continuous at point $c$ also. $f/g$ is continuous, provided $g(c)\ne 0$.
2. Boundedness: If $f$ is continuous on the closed bounded interval $[a,b]$, then there is a number $K$ such that $|f(x)|\le K$ for each $x$ in $[a,b]$.
3. Max/Min: If $f$ is continuous on the closed bounded interval $[a,b]$, then $f$ has a maximum and a minimum on $[a,b]$. They may be located at the end points.


## Sets

__Interior Point__: The point $\bf x$ is an interior point of the set $S$ if $\bf x$ is in $S$ and if there is some $\epsilon$-ball around $\bf x$ that contains only points in $S$.   The __interior__ of $S$ is the collection of all interior points in $S$.  The interior can also be defined as the union of all open sets in $S$.

* If the set $S$ is circular, the interior points are everything inside of the circle, but not on the circle's rim.
* Example: The interior of the set $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2< 4 \}$ .
	
__Boundary Point__: The point $\bf x$ is a boundary point of the set $S$ if every $\epsilon$-ball around $\bf x$ contains both points that are in $S$ and points that are outside $S$.  The __boundary__ is the collection of all boundary points.

* If the set $S$ is circular, the boundary points are everything on the circle's rim.
* Example: The boundary of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 = 4 \}$.
	
__Open__: A set $S$ is open if for each point $\bf x$ in $S$, there exists an open $\epsilon$-ball around $\bf x$ completely contained in $S$.

* If the set $S$ is circular and open, the points contained within the set get infinitely close to the circle's rim, but do not touch it.
* Example: $\{ (x,y) : x^2+y^2<4 \}$
	
__Closed__: A set $S$ is closed if it contains all of its boundary points.

* If the set $S$ is circular and closed, the set contains all points within the rim as well as the rim itself.
* Example: $\{ (x,y) : x^2+y^2\le 4 \}$
* Note: a set may be neither open nor closed. Example: $\{ (x,y) : 2 < x^2+y^2\le 4 \}$
	
__Complement__: The complement of set $S$ is everything outside of $S$.

* If the set $S$ is circular, the complement of $S$ is everything outside of the circle.
* Example: The complement of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 > 4 \}$.
	
__Closure__: The closure of set $S$ is the smallest closed set that contains $S$. 

* Example: The closure of $\{ (x,y) : x^2+y^2<4 \}$ is $\{ (x,y) : x^2+y^2\le 4 \}$
	
__Bounded__: A set $S$ is bounded if it can be contained within an $\epsilon$-ball.

* Examples:  Bounded: any interval that doesn't have $\infty$ or $-\infty$ as endpoints; any disk in a plane with finite radius.
* Unbounded: the set of integers in ${\bf R}^1$; any ray.
	
__Compact__: A set is compact if and only if it is both closed and bounded.
	
__Empty__: The empty (or null) set is a unique set that has no elements, denoted by \{\} or $\o$. 

* Examples: The set of squares with 5 sides; the set of countries south of the South Pole.
* The set, $S$, denoted by $\{\o\}$ is technically \emph{not} empty. That is because this set contains the empty set within it, so $S$ is not empty.