Functions, Limits, and Continuity
Terms
Composition
-
Given two functions
f : S→T
and
g : T→U
, the function
g
o
f : S→U
defined by applying
f
and then
g
is called the
composition of
g
and
f
.
Continuous
-
A function
f
is said to be continuous at a point
x
0
of its domain if
it has a limit there, and that limit agrees with the value
f (x
0)
. In
mathematical notation:
If the function is continuous at every point of its domain, then one simply says it is continuous.
f (x) = f (x
0)
|
If the function is continuous at every point of its domain, then one simply says it is continuous.
Domain
-
The set of values that a function
f
takes as its input.
Elementary Functions
-
A standard library of functions including the linear, polynomial, rational, power, and
trigonometric functions.
Function
-
A rule
f
that assigns to each element in a set
S
a unique element in a set
T
, which is written
f : S→T
.
Image
-
The value
f (x)
which a function
f
assigns to a particular value
x
in its domain.
Interval
-
A set consisting of the real numbers between two fixed points, possibly
including one or both of these endpoints. An open interval
(a, b)
is
the set of real numbers
x
such that
a < x < b
(excluding the endpoints). A
closed interval
[a, b]
is the set of real numbers
x
such that
a≤x≤b
(including the endpoints).
Invertible
-
A term that describes a function
f : S→T
such that there exists a function
g : T→S
with
(g
o
f )(x) = x
for each element
xâààS
.
Limit
-
The value that a function
f (x)
approaches as
x
approaches a particular value
x
0
. This is the intuition behind a more rigorous
definition.
Linear Function
-
A function of the form
f (x) = ax + b
, where
a
and
b
are real numbers. The graph of
such a function is a straight line.
Polynomial Function
-
A function of the form
f (x) = a
n
x
n + ... + a
1
x + a
0
for real numbers
a
0,…a
n
.
Power Function
-
A function of the form
f (t) = Cr
t
that is used to model exponential growth or decay.
Logarithm
-
The inverse of a power function
f (t) = Cr
t
is the logarithm with base
r
, denoted
logr(t)
.
Range
-
The set within which the output of a function
f
lies.
Rational Function
-
A function that is formed by taking the quotient of two polynomials.
Set
-
A collection of objects (which are called elements).
Slope
-
The number
a
for the linear function
f (x) = ax + b
, indicating the steepness of the graph
of
f
.
Trigonometric Function
-
A periodic function involving sines, cosines, tangents, or their reciprocals or inverses.
y
-intercept
-
The number
b = f (0)
for a linear function
f (x) = ax + b
, indicating the vertical coordinate
of the intersection point of the graph of
f
with the
y
-axis.
