The operation of combining two functions by
applying them one after the other. That is, the composition of f(x)
and g(x), denoted by , is defined by
the equation .
Either the exponent of a power function or the highest power in a polynomial
function. For example, the degree of f(x)
= -4x5 is 5, while the degree of h(x)
= 5 - x + 4x3 - 7x2 is 3.
We say that a function, f, is even, if it satisfies the equation f(-x) = f(x)
for all values of x. This terminology is based on the fact
that all even power functions (e.g., x2, x4,
x-6, etc.) satisfy this equation.
A relationship between two sets of values, where
the second value
(the "output") is
uniquely determined (according to some clearly specified
"rule") by the first value
(the "input").
The notation used to display the name of a function
and an associated input value, expression, or independentvariable, directly to its right in parentheses
(e.g., f(1) or f(2t +1) or f(x)).
A pictorial representation of a numerical relation,
where each ordered pair in the relation corresponds to a point in the
plane, with the two numbers corresponding to the horizontal and vertical
displacement from the origin, respectively.
Any text in a Web (HTML) document that is colored and underlined, which, when selected
with the mouse in a Web-browser, will cause the browser to move to a specific location in
an indicated document. This allows the author to guide the reader appropriately through a
collection of documents.
A function, g(x),
is the inverse of another function, f(x),
if they are reverse relations. Equivalently, they are inverses if
they "cancel" each other under composition, that is, f(g(x))
= x and g(f(x)) = x.
A function, f(x),
is invertible if there is a function, g(x),
which is its inverse. In this case, there
is only one possible inverse, and we denote it
by f -1(x). This inverse
is characterized by the equations, f(f -1(x))
= x and f -1(f(x)) = x.
We say that a function, f, is odd, if it satisfies the equation f(-x) = -f(x)
for all values of x. This terminology is based on the fact
that all odd power functions (e.g., x3, x5,
x-1, etc.) satisfy this equation.
The period of a function, f, is
the length of the shortest interval over which it repeats its
values. More precisely it is the smallest number p such
that f(x + p) = f(x) for all inputs x
such that x + p and x are both in the domain of f.
A function that is a sum of power functions,
with positive integer exponents, multiplied by constants, such as f(x) = x3 - 2x2
+ 4x + 7 = 1x3 + -2x2
+ 4x + 7x0.
A radical is a symbol traditionally used to denote square
roots, cube roots, etc. A radical function is
the inverse of a (restricted) power function with
positive integer power.
A rational number can be written as a ratio (i.e., fraction) of
two integers. A rational function is
one whose formula can be given as the ratio (i.e., quotient) of two polynomials.
A transformation of the x-y plane that moves each point to the opposite side of a fixed
line, called its axis, along the line perpendicular to the axis so that the resulting
point is the same distance from the axis as the original point.
A relation is a collection of ordered pairs, like (father, son).
If these pairs are real numbers, like (5, 2), then the relation may be
graphed on a pair of coordinate axes, and they may satisfy some
formula, but this is not necessary. Otherwise, we may give an
alternative graphical description, using an arrow diagram.
A root of a polynomial function, f,
is just a solution to the equation f(x) = 0, that is an
input that yields an output of 0. Geometrically, this corresponds
to the point (x, 0) on the horizontal axis where the graph of f
crosses.
This is error introduced into a calculation by rounding off the
results of intermediate steps to values which are close, but not exactly
correct. The more times this is done in a calculation, in general,
the more inaccurate the final result will be. Note: This
means that, when using a calculator, it is best to use algebra to obtain
an exact expression for the desired value, and enter the entire
expression (with proper parenthesis) into the calculator all at once.
A transformation of the number line or plane that takes each point the same fraction towards or
away from the origin in a given direction (e.g., stretching or shrinking).
The slope of a line is the ratio, when we move between any two
points on the line, of the change in vertical position over the change
in horizontal position.
A solution to an equation is a
choice of value for each variable so that, when the values are substituted for the
variables, both sides of the equation simplify to the same value.
A subscript
is a notation that we attach to the lower right of a constant, variable
or function name to create a new, but similarly named constant, variable
or function. For example, in x1, loge,
and reffective, the number 1, the constant e,
and the word "effective" all serve as subscripts to the
variable name, x, the function name, log, and the constant name,
r.
A letter that is used to represent a numerical quantity whose value
may or may not yet be known and may or may not be constant.
It may not even have a specific value, but rather be allowed to range
over a set of possible values.
Send
questions or comments to jwicks@northpark.edu