Properties of Functions:
Definition of a Function: A function is a rule
or formula that associates each element in the set X (an input)
to exactly one and only one element in the set Y (the output).
Different elements in X can have the same output, and not every
element in Y has to be an output.
Definition of the Domain of a Function: The set
of all possible inputs of a function is defined as the domain.
The domain of a realvalued function defined by a formula for y in
terms of x will be the set of all x inputvalues that result in a real
y outputvalue unless the domain of the function is further
restricted.
Definition of the Range of a Function: The set
of all possible outputs of a function is defined as the range.
The range of a realvalued function defined by a formula for y in
terms of x will be the set of all y outputvalues that result from the
x inputvalues in the domain.
Function Notation: Given that f(x) is given by
some formula containing x, f(B) will be the same formula with each x
replaced by B.
Linear Function Definition: If a function may
be written in the form f(x) = mx + b where x is the independent
variables and m and b are constants, then f(x) represents a linear
function. The variable m is defined as the slope and the point (0, b)
represents the yintercept. An equation in this form is known to be in
SlopeIntercept Form.
Linear Function Slope Definition: Given that
f(x) = mx + b, then m is defined as the slope where:
for any two points (x_{1}, y_{1}) and (x_{2},
y_{2}) on the line. Graphically, the slope represents
the change in y with respect to x on the graph of the line.
Linear Functions of Parallel Lines If two
linear functions are given by f(x) = m_{1}x + b and g(x) = m_{2}x
+ b, and m_{1} = m_{2}, then the graphs of f(x) and
g(x) will consist of two lines that are parallel to each other.
Linear Functions of Perpendicular Lines If two
linear functions are given by f(x) = m_{1}x + b and g(x) = m_{2}x
+ b, and m_{1} = 1/m_{2}, then the graphs of f(x) and
g(x) will consist of two lines that are parallel to each other.
Graphs of Even Functions Given a function f(x),
if f(c) = f(c) for all c in the domain, then f(x)
is an even function and its graph will have symmetry with
respect to the yaxis.
Graphs of Odd Functions Given a function f(x),
if f(c) = f(c) for all c in the domain, then
f(x) is called an odd function and its graph will have symmetry
with respect to the origin. Symmetry with respect to the origin
implies that a 180 degree rotation of the graph about (0,0) results in
an identical graph.
Functions Shifted Left Given a function f(x)
and its graph and a value of c>0, the graph of f(x + c) will be a
shift of the graph of f(x) left by "c" units. This is known as the
Left Shift Function Rule.
Functions Shifted Right Given a function f(x)
and its graph and a value of c>0, the graph of f(x  c) will be a
shift of the graph of f(x) right by "c" units. This is known as
the Right Shift Function Rule.
Functions Shifted Up Given a function f(x) and
its graph and a value of c>0, the graph of f(x) + c will be a
shift of the graph of f(x) up by "c" units. This is known as the
Vertical Shift Function Rule.
Functions Shifted Down Given a function f(x)
and its graph and a value of c>0, the graph of f(x)  c
will be a shift of the graph of f(x) down by "c" units. This is known
as the Vertical Shift Function Rule.
Function Reflected Across Xaxis Given a
function f(x) and its graph, the graph of g(x) = f(x) will be a
reflection of the graph of f(x) across the xaxis. This is known
as the Xaxis Reflection Function Rule.
Function Reflected Across Yaxis Given a
function f(x) and its graph, the graph of g(x) = f(x) will be a
reflection of the graph of f(x) across the xaxis. This is known
as the Yaxis Reflection Function Rule.
Function Vertically Stretched Or Shrunk Given
a function f(x) and its graph and a value of c>0, the graph of
g(x) = c●f(x) will be a vertical stretch of the graph of f(x).
This means that all yvalues of g(x) will be equal to c times
the respective yvalues of f(x). This is known as the Vertical
Stretch Function Rule.
Definition of a Polynomial Function If f(x)
may be written in the form a_{1}x^{n} + a_{2}x^{n1}
+ a_{3}x^{n2} + .... + a_{n}, then f(x) is a
polynomial function of degree n where a_{1}, a_{2},
... a_{n} are real coefficients. Linear functions are
1st degree polynomials and quadratic functions are 2nd degree
polynomials.
Graphs of Polynomials Given a function f(x) is
a polynomial, it's xintercepts will be located at the xvalues x=c
such that f(c) = 0. Other solution points on the graph will be
located between each two xintercepts.
Standard Form of Quadratic Functions Quadratic
functions of the form f(x) = ax^{2} + bx + c may always be
rewritten in the form y = a(x  h)^{2} + k.
Function shift rules may then be applied to state that the graph will
be a vertical stretch of y = x^{2} and will be shifted right,
left, up, or down according to the values of h and k.
Graphs of Quadratic Functions in Form f(x) = ax^{2}
+ bx + c Given f(x) = ax^{2} + bx + c, the graph will be a
shift of g(x) = ax^{2 }(meaning it has the same shape), and
will have a vertex at x=b/2a, y = f(b/2a).
Property of The Vertex of a Quadratic Function
The vertex of f(x) = ax^{2} + bx + c will be the lowest point
of the graph if a>0 and will be the highest point of the graph if a<0.
The vertex represents the minimum value of the function for a>0 and
represents the maximum value of the function if a<0.
Function Operations Given two functions f(x)
and g(x), the operations (f+g)(x), (fg)(x), (fg)(x), and (f/g)(x) are
defined in the following way:
 (f + g)(x) = f(x) + g(x)
and is sometimes denoted f+g
 (f  g)(x) = f(x)  g(x)
and is sometimes denoted f  g
 (fg)(x) = f(x)● g(x)
and is sometimes denoted fg
 (f/g)(x) = f(x)/g(x)
provided g(x)≠0.
This is sometimes denoted f/g
Function Composition Given two functions f(x)
and g(x), the function compostion (fog)(x), is defined in the
following way:
(f o g)(x) = f [g(x)] and
is sometimes denoted as f o g
In essence, composition implies that you input the entire
formula of the second function in for each xvalue of the the formula
in the first function, assuming x is the variable used.
Definition of Inverse Functions Given two
functions f(x) and g(x), if (f o g)(x) = x and (g o f)(x) = x, then
f(x) is the inverse of g(x) and g(x) is the inverse of f(x). Each of
these functions reverses the operations of the other function in
reverse order. In that sense, the inverse of f(x) will consist
of the identical formula with x and y interchanged  the solution for
y results in "reversing" all operations on x and thus results in the
formula for the inverse function.
We denote the inverse of f(x) as f^{ 1}(x) and we
denote the inverse of g(x) as g^{1}(x).
Domain and Range of Functions That Are Inverses of
Each Other Given two functions f(x) and g(x) are inverses of each
other, then
The Domain of f(x) will consist of the same interval as the
Range of g(x).
The Range of f(x) will consist of the same interval as the
Domain of g(x).
OneToOne Requirement For f(x) To Have an Inverse
Function Given a function f(x), it will only have an inverse if
and only if each yvalue in it's range corresponds to only 1 xvalue
in it's specified domain. When this is the case that each y is
obtained from only 1 xvalue we say f(x) is a onetoone function.
Note that a graphical way to determine that f(x) is not
onetoone is to show that a horizontal line passes through more than
1 point. This is often referred to as the Horizontal Line Test.
