Week 2-3 - Functions & Limits #

Functions #

Deﬁnition: A function f is a rule that assigns to each element x in a set D exactly one element, called $f(x)$ , in a set E.
- The set D is called the domain of the function.
- The number $f(x)$ is the value of f at x and is read “f of x”.
- The range of f is the set of all possible values of $f(x)$ as x varies throughout the domain.
- A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable.
Four ways to represent a functions
- verbally (by a description in words)
- numerically (by a table of values)
- visually (by a graph)
- algebraically (by an explicit formula)
A catalog of essential functions
- linear function
  - $f(x) = mx+b$
- polynomials
  - $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+…+a_{2}x^{2}+a_{1}x^{1}+a_{0}$
  - The domain of any polynomial is $\mathbb{R} = (-\infty, + \infty)$ .
  - If the leading coefficient $a_{n}\ne 0$ , then the degree of the polynomial is n.
    - n=2: quadratic function
    - n=3: cubic function
- power function
  - $f(x) = x^a$
    - a = n, where n is a positive integer
    - a = 1/n, where n is a positive integer. It’s a root function.
    - a = -1: reciprocal function
- rational function
  - A rational function f is a ratio of two polynomials:
    - $f(x)=\frac{P(x)}{Q(x)}$ , ${Q(x)\ne0}$
    - $f(x)=\frac{2x^{4}-x^{2}+1}{x^{2}-4}$ , ${x|x \ne \pm 2}$ :
- algebraic function
  - A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots)
- trigonometric function
  - $f(x)=\sin{x}$
- exponential function
  - $f(x)=b^x$
  - law of exponential function:
    - $b^{x+y}=b^x+b^y$
    - $b^{x-y}=\frac{b^x}{b^y}$
    - $(b^{x})^{y}=b^{xy}$
    - ${ab}^{x}=a^{x}b^{x}$
- logarithmic function
  - $f(x)=\log_{b}{x}$

Combinations of Functions #

$(f+g)(x)=f(x)+g(x)$
$(f-g)(x)=f(x)-g(x)$
$(fg)(x)=f(x)g(x)$ , the domain of fg is $A \cap B$
$(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$ , the domain of f/g is ${x \in A \cap B\\ |\\ g(x) \ne 0}$ .
$(f \circ g)(x) = f(g(x))$
- composition (or composite) of f and g, denoted by $f \circ g$ (“f circle g”).

Inverse of Functions #

Definition:
- If a function maps every input to exactly one output, an inverse of that function maps every “output” to exactly one “input.”
denoted by $f^{-1}$ , and read “f inverse”.
to function $N=f(t)$ , the inverse function will be $t=f^{-1}(N)$ .

one-to-one functions #

A function is one-to-one if for every value in the range(f(x)), there is exactly one value in the domain(x).
- domain of $f^{-1}$ = range of $f$
- range of $f^{-1}$ = domain of $f$
- for example: $f(x)=x^3$ is a one-to-one function, $f(x)=x^2$ is not.

Limits #

Definition:
- $\displaystyle\lim_{x \to a}{f(x)}=L$
- the limit of f(x), as x approaches a, equals L
- if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by restricting x to be sufficiently close to a (on either side of a) but NOT equal to a.(This means that in finding the limit of f(x) as x approaches a, we never consider x = a.)

Limit Laws #

Suppose that c is a constant and the limits $\displaystyle\lim_{x \to a}{f(x)}$ and $\displaystyle\lim_{x \to a}{g(x)}$ exist, Then:
1. $\displaystyle\lim_{x \to a}{[f(x)+g(x)]}=\lim_{x \to a}{f(x)}+\lim_{x \to a}{g(x)}$
2. $\displaystyle\lim_{x \to a}{[f(x)-g(x)]}=\lim_{x \to a}{f(x)}-\lim_{x \to a}{g(x)}$
3. $\displaystyle\lim_{x \to a}{cf(x)}=c\lim_{x \to a}{f(x)}$
4. $\displaystyle\lim_{x \to a}{[f(x)g(x)]}=\lim_{x \to a}{f(x)}\cdot \lim_{x \to a}{g(x)}$
5. $\displaystyle\lim_{x \to a}{\frac{f(x)}{g(x)}}=\frac{\displaystyle\lim_{x \to a}{f(x)}}{\displaystyle\lim_{x \to a}{g(x)}}$ , if $\displaystyle\lim_{x \to a}{g(x)} \ne 0$
These five laws can be stated verbally as follows:
1. Sum Law : The limit of a sum is the sum of the limits.
2. Difference Law : The limit of a difference is the difference of the limits.
3. Constant Multiple Law : The limit of a constant times a function is the constant times the limit of the function.
4. Product Law : The limit of a product is the product of the limits.
5. Quotient Law : The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).

Squeeze Theorem #

if $g(x) \le f(x) \le h(x)$ and $\displaystyle\lim_{x\to{a}}g(x) = \lim_{x\to{a}}h(x) = L$ , then $\displaystyle\lim_{x\to{a}}f(x)=L$
Sample: to prove $\displaystyle\lim_{x\to{0}}\frac{\sin(x)}{x}=1$
- we know:
  - $\displaystyle\lim_{x\to{0}}\cos(x)=1=\lim_{x\to{0}}1$
  - $\cos(x) \le \frac{\sin(x)}{x} \le 1$
    - to prove this(from www.khanacademy.org)
- then:
  - $\displaystyle\lim_{x\to{0}}\cos(x) \le \lim_{x\to{0}}\frac{\sin(x)}{x} \le 1$
- so:
  - $\displaystyle\lim_{x\to{0}}\frac{\sin(x)}{x}=1$

Continuity #

Definition 1:
- A function f is continuous at a number a, if $\displaystyle\lim_{x \to a}f(x)=f(a)$
  - Notice that this Definition implicitly requires three things if f is continuous at a: 1. f(a) is defined (that is, a is in the domain of f) 2. $\displaystyle\lim_{x \to a}f(x)$ exists 3. $\displaystyle\lim_{x \to a}f(x) = f(a)$
Definition 2:
- A function f is continuous from the right at a number a if $\displaystyle\lim_{x \to a^{+}}f(x) = f(a)$ , and f is continuous from the left at a if $\displaystyle\lim_{x \to a^{-}}f(x) = f(a)$
Definition 3:
- A function f is continuous on the interval (a, b), if for all points c so that a < c < b, f(x) is continuous at c.
  - close intervals:
    - To say “f(x) is continuous on the interval [a, b]”, means:
      - f(x) is continuous on the interval (a, b)
      - $\displaystyle\lim_{x \to a^{+}}f(x) = f(a)$
      - $\displaystyle\lim_{x \to b^{-}}f(x) = f(b)$

The Intermediate Value Theorem #

Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where $f(a) \ne f(b)$ . Then there exists a number c in [a, b] such that $f(c) = N$ .
For example: how to approximate root two?
- use function $f(x)=x^2-2$
- when $x=\sqrt{2}$ , this function equals 0,
- so i just need to look for a positive value, that I can plug into this function to make it equals zero.
- We know that f(1)=-1<0 and f(2)=2>0. So base on The Intermediate Value Theorem, there must be a value in domain (1, 2) that exist c such that f(c) = 0.
- to continue calculating f(1.5)=0.25>0, we got $c \in (1, 1.5)$ , then $c \in (1.4, 1.5)$ … then we are getting closer and closer to $\sqrt{2}$ .

Fixed Point #

Definition
- A fixed point of a function f is a number c in its domain such that f(c) = c. (The function doesn’t move c; it stays fixed.)
  - if f(x) continuous on [0,1], and $0 \le f(x) \le 1$ , then there is an x in domain [0, 1], exist f(x) = x.
    - To Prove:
    - Assumption: g(x) = f(x) - x, so g(x) is continuous
    - g(0) = f(0) - 0 >= 0
    - g(1) = f(1) - 1 <= 1
    - Base on the IVT(Intermediate Value Theorem), there must be an x such that g(x) = 0, which is f(x) = x.

Infinity #

Definition
- $\displaystyle\lim_{x \to a}f(x) = \infty$ means that f(x) is as large as you like provides x is close enough to a.
  - $\displaystyle\lim_{x \to \pi/2}\tan(x) = \infty$
- $\displaystyle\lim_{x \to \infty}f(x) = L$ means that f(x) is close enough to L provided x is large enough.
  - $\displaystyle\lim_{x \to \infty}\tan^{-1}(x) = \frac{\pi}{2}$
- if $\displaystyle\lim_{x \to \infty}f(x) = L$ or $\displaystyle\lim_{x \to -\infty}f(x) = L$ , then the line y = L is called a horizontal asymptote of the curve y = f(x) :
Potential Infinity vs Actual Infinity (from Wikipedia)
- Actual Infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality;
  - Such as the set of all natural numbers, an infinite sequence of rational numbers.
- Potential Infinity is a non-terminating process (such as “add 1 to the previous number”) produces an unending “infinite” sequence of results, but each individual result is finite and is achieved in a finite number of steps.
Precise Definitions
- $\displaystyle\lim_{x \to a}f(x) = L$ means:
  - for all $\epsilon > 0$ , there is $\delta>0$ ,
  - so that if $0 < |x - a| < \delta$ ( $x \ne a$ and x in within $\delta$ of a), then $|f(x)-L| < \epsilon$ (f(x) is within $\epsilon$ of L).
    - |x - a| is the distance from x to a and |f(x) - L| is the distance from f(x) to L.
    - so $\displaystyle\lim_{x \to a}f(x) = L$ means that the distance between f(x) and L can be made arbitrarily small by requiring that the distance from x to a be sufficiently small (but not 0).
- For Example: $\displaystyle\lim_{x \to 2}x^2 = 4$
  - Let’s say $\epsilon = 0.1$ , that means $|f(x)-4| < 0.1$ , 3.9 < f(x) < 4.1,
  - Base on the definition, there should be a $\delta$ , that $2 - \delta < x < 2 + \delta$ to satisfy the demand.
  - Try $\delta = 0.01$ . We got 1.99 < x < 2.01, 3.9601 < x^2 < 4.0401, 3.9 < f(x) < 4.1 which suit the demand.
- Another Example: $\displaystyle\lim_{x \to 10}2x = 20$
  - Let $\epsilon > 0, \delta = \epsilon / 2$
  - if $0<|x-10|<\delta$ , then,
  - $0<2|x-10|<2\delta=\epsilon$ , and so,
  - $0<|2x-20|<\epsilon$

Review Questions #

defintion of function and inverse of function. what are the notations?
- double check the definition of inverse of function.
what is linear/polynomial/power/rational function?
- what is the difference between polynomial and power function?
- what is the relation between polynomial and rational function?
what is algebraic/trigonometric/expoential/logarithmic function?
what are the limit and limit laws?
- sum/difference/constant mutiple/product/quotient law
what is sqeeze theorem?
how to check whether a function is countinuous at a point?
what is the intermediate value theorem?
wthat is a fixed point?
what is infinity?
- check the precise definition.
[optional]what is the difference between actual infinity and potential infinity?