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
.
D
is called the domain of the function.f
at x
and is read “f of x”.f
is the set of all possible values of $f(x)$ as x
varies throughout the domain.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
A catalog of essential functions
linear function
polynomials
n=2
: quadratic function
n=3
: cubic function
power function
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
f
is a ratio of two polynomials:
algebraic 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
exponential function
logarithmic function
fg
is $A \cap B$f/g
is $\{x \in A \cap B\ |\ g(x) \ne 0\}$.f
and g
, denoted by $f \circ g$ (“f circle g”).f(x)
), there is exactly one value in the domain(x
).
f(x)
, as x
approaches a
, equals L
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
.)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:
These five laws can be stated verbally as follows:
f
is continuous at a number a, if $\displaystyle\lim_{x \to a}f(x)=f(a)$
f
is continuous at a
:
f(a)
is defined (that is, a
is in the domain of f
)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 $\displaystyle\lim_{x \to a^{-}}f(x) = f(a)$f
is continuous on the interval (a, b)
, if for all points c
so that a < c < b
, f(x)
is continuous at c
.
f(x)
is continuous on the interval [a, b]
", means:
f(x)
is continuous on the interval (a, b)
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 $f(c) = N$.
0
,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
.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 $\sqrt{2}$.f
is a number c
in its domain such that f(c) = c
. (The function doesn’t move c
; it stays fixed.)
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
.
g(x) = f(x) - x
, so g(x)
is continuousg(0) = f(0) - 0 >= 0
g(1) = f(1) - 1 <= 1
x
such that g(x) = 0
, which is f(x) = x
.Definition
f(x)
is as large as you like provides x
is close enough to a
.
f(x)
is close enough to L
provided x
is large enough.
y = L
is called a horizontal asymptote of the curve y = f(x)
:
Potential Infinity vs Actual Infinity (from Wikipedia)
Precise Definitions
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.3.9 < f(x) < 4.1
,1.99 < x < 2.01
, 3.9601 < x^2 < 4.0401
, 3.9 < f(x) < 4.1
which suit the demand.