Precalculus: Functions
Terms
Cartesian Product
-
The set of all possible ordered pairs
(a, b)
composed of elements taken from
the two sets,
A
and
B
.
Composition
-
A function operation symbolized
(f
o
g)(x)
that is equivalent to
f (g(x))
.
Defined
-
A function is defined at a given value of the independent variable if it
assigns that input an output; defined means "takes on a value".
Dependent Variable
-
The output variable of a function; the variable whose value depends on the
input, or independent variable.
Domain
-
The set of all inputs for which a function or relation is defined.
Even Function
-
A function
f
is even if
f (x) = f (- x)
.
Function
-
A relation which assigns exactly one element in its range for each
element in its domain.
Horizontal Line Test
-
The test by which it is shown whether a function is a one-to-one
function or not, and therefore whether its inverse is a function.
Independent Variable
-
The variable of a function which does not depend on the other variable -- it
is the input.
Inverse
-
A relation which assigns a correspondence from the elements of the range
to those of the domain. The inverse of a function or relation can be found
by interchanging the variables in the function or relation.
Odd Function
-
A function
f
is odd if
f (x) = - f (- x)
.
One-to-One Function
-
A function is one-to-one if each element in its range is paired with
exactly one element from its domain.
Periodic Function
-
A function is periodic if and only if
f (x) = f (x + c)
, for all values
x
,
where
c
is a constant. A periodic function repeats itself at regular
intervals.
Piecewise Function
-
A function is piecewise if and only if it uses different rules for different
parts of its domain.
Range
-
The set of all outputs of a function or relation.
Relation
-
A rule that associates the elements of one set with those of another set. A
relation can also be thought of as all of the ordered pairs which satisfy
the rule.
Undefined
-
A function is undefined at a given value of its independent variable if
for that value, there is no output--this occurs when a particular input
creates a situation in which there is division by zero, or an even root of a
negative number, for example.
Vertical Line Test
-
The test by which a relation is either shown to be a function or not.
The graph of a function does not intersect with a vertical line more than once.
