Functions, Limits, Continuity
Terms
Closed Interval
-
A set of numbers on the number line that is bounded by two endpoints and that includes
the endpoints. For example, the closed interval
[- 2, 2]
contains all the numbers greater
than or equal to -2 and less than or equal to 2. A closed endpoint is denoted by a bracket
around the endpoint. Intervals may also be closed at one endpoint and open at the
other.
Composite Function
-
A combination of two functions in which the output of one function is the input for
the other. The composite of
f
and
g
, written as
(f
o
g)(x)
,
means
f (g(x))
.
Constant Function
-
This is a function whose value is always constant and does not vary with the input.
For example,
f (x) = 4
is a constant function.
Continuous
-
Intuitively, a function is continuous if you can draw it without lifting your pen from
the paper.
Formally, a function
f (x)
is continuous at a point
x = c
if the following is
true at that point:
A continuous function is one that is continuous for all points in its domain.
f (x) = f (c)
|
A continuous function is one that is continuous for all points in its domain.
Domain
-
The domain of a function
f
is the set of all real numbers for which
f
is defined.
Even Function
-
A function for which
f (- x) = f (x)
for all
x
in the domain. This function is
symmetric with respect to the
y
-axis.
Function
-
A rule which assigns to each element
x
in the domain a single element
y
in the
range.
Horizontal Line Test
-
A graphical test to determine whether a function can be considered a one-to-one
function. If no horizontal line drawn on the graph of the function passes through more
than one point, then the function is a one-to-one function.
Intermediate Value Theorem
-
If
f
is a continuous function on a closed interval
[a, b]
, then for every value
r
that lies between
f (a)
and
f (b)
, there exists a constant
c
on
(a, b)
such that
f (c) = r
.
Interval Notation
-
A convenient way of representing sets of numbers on a number line bound by two
endpoints. See closed interval and open interval.
Left-Hand Limit
-
This is the one-sided limit obtained by allowing the variable
x
to approach the
constant
c
from "the left side" only, i.e. from values of
x
less than
c
.
Limit
-
This is the single value that a function
f (x)
approaches as the variable
x
approaches a
constant
c
. Ordinarily, the term "limit" used by itself refers to a two-sided limit.
Linear Function
-
This is a polynomial function of the first degree. The variable
x
is only raised to
the first power. The graph of this function is always a straight line. The function is of
the form
f (x) = ax + b
where
a
and
b
are constants.
Odd Function
-
This is a function
f
for which
f (- x) = - f (x)
for all
x
in the domain. The graph of this
function is symmetric with respect to the origin.
One-Sided Limit
-
This is the sort of limit that is obtained when the variable
x
is allowed to approach
the constant
c
from only one side, i.e. from values greater than
c
or values less than
c
, but not both. One-sided limits can be either a left-hand limit or right-hand
limit.
One-to-One Function
-
This is a type of function that assigns a different element in the range to each
element in the domain so that no two domain elements map to the same range element. A
graphical way to test for a one-to-one function is to perform the horizontal line test.
Open Interval
-
A set of numbers on the number line that is bounded by two endpoints and that does not
include the endpoints. For example, the open interval
(- 2, 2)
contains all the numbers
greater than -2 and less than 2, but does not include
-2
and
2
themselves. An open
endpoint is denoted by a parenthesis around the endpoint. Intervals may also be open at
one endpoint and closed at the other.
Piecewise-Defined Function
-
A function that is defined differently for different intervals in its domain.
Polynomial Function
-
Any function of the form
where a 0, a 1, a 2,...a n are constants and n is a nonnegative integer. n denotes the "degree" of the polynomial. Examples of polynomial functions of varying degrees include constant functions, linear functions, and quadratic functions.
| f (x) = a 0 + a 1 x + a 2 x 2 + ....a n-1 x n-1 + a n x n |
where a 0, a 1, a 2,...a n are constants and n is a nonnegative integer. n denotes the "degree" of the polynomial. Examples of polynomial functions of varying degrees include constant functions, linear functions, and quadratic functions.
Quadratic Function
-
A polynomial function of the second degree. The highest power that the variable
x
is raised to is the second power. These functions are of the form
f (x) = ax
2 + bx + c
where
a
,
b
, and
c
are constants.
Range
-
This is the set of all possible outputs for the function
f
.
Rational Function
-
This is a function of the form
where f and g are both polynomial functions.
r(x) = 
|
where f and g are both polynomial functions.
Right-Hand Limit
-
This is the one-sided limit obtained by allowing the variable
x
to approach the
constant
c
from "the right side" only, i.e. from values of
x
greater than
c
.
Squeeze Rule
-
A method for finding the limit of a function
h(x)
:
Suppose
f (x)≤h(x)≤g(x)
for all
x
in an open
interval containing
c
(except possibly at
c
itself).
If
then
h(x)
exists, and
h(x) = L
.
|
f (x) =
g(x) = L
|
then
h(x)
exists, and
h(x) = L
.
Two-sided Limit
-
A kind of limit in which
x
is allowed to approach
c
from values less than
c
and values greater than
c
with the exact same result. Thus, the two-sided limit exists
only when both one-sided limits exist and are equal.
Vertical Line Test
-
A graphical test used to determine whether a rule is a function. If we cannot draw a
vertical line through more than one point on a graph, then that graph represents a
function.
