Download Algebraic Functions

This section is a field guide to all of the
functions with which we must be proficient
in a Calculus course.
An elementary function is one built from certain basic
elements using certain allowed operations.
An algebraic function is one using only the operations
of +, -, *, /, and powers or radicals.
Transcendental functions include exponential,
logarithmic, and trigonometric functions.
A polynomial is an algebraic function that can be written
as p ( x ) = a0 + a1 x + a2 x 2 +... + an x n
Each ak is called a coefficient, and can have any real
number value.
The degree of a polynomial is the largest exponent for
which the coefficient is not 0.
Polynomial graphs are “smooth” everywhere – they
have no “sharp points”.
Polynomial graphs have no “breaks” in them – they
are continuous everywhere.
The domain (set of valid input values) of a polynomial
is all real numbers, which is (-∞, ∞).
The range (set of outputs produced) of a polynomial
varies with its degree.
Degree = 0
Degree = nonzero, even
Degree = odd
Degree = 0  range = {#}
Degree = even (≠ 0)  range = (-∞, max] or [min, ∞)
Degree = odd  range = (-∞, ∞)
A rational function is a function that can be written as:
p ( x)
f ( x) =
q ( x)
Here, both p and q are polynomials.
The domain is { x : q ( x ) ¹ 0}; the range varies a lot from
function to function.
For any x value for which q(x) = 0 but p(x) ≠ 0, the
rational function f has a vertical asymptote.
As
x ® c, y ® ±¥ means x = c is a V.A.
f ( x) =
2x
x-3
A function of the form f ( x ) = n g ( x ) is called a radical
function.
The “inside” function, g(x), is called the radicand.
The index of the radical is n. For a square-root
function, the index is 2 even though it is not written in
the radical notation.
Domain of f ( x ) = n g ( x ) :
n even  {x: g(x) ≥ 0 and g(x) is defined}
n odd  {x: g(x) is defined}
f ( x) = x - 2
f ( x) = 3 x
A function of the form f ( x ) = b x where b > 0 is an
exponential function.
If b <1, the function is decreasing. If b > 1, the
function is increasing.
f ( x) = ex
f ( x ) = 0.5x
The domain is all real numbers: (-∞, ∞).
The range is all positive numbers: (0, ∞).
The point (0, 1) is on every bx curve.
The natural exponential function is ex. This function
has many nice calculus properties.
The logarithm function with base b (where b > 0),
f ( x ) = logb ( x ) is the inverse of the exponential function
defined by f ( x ) = b x .
This means y = logb ( x ) iff b y = x
If b > 1, the function increases and if b < 1 the
function decreases.
f ( x ) = log0.5 ( x )
Domain = all positive numbers = (0, ∞)
Range = all real numbers = (-∞, ∞)
f ( x ) = log3 ( x )
The base 10 logarithm is called the common logarithm and is
denoted as log(x).
The base e logarithm is called the natural logarithm and is denoted
as ln(x).
All logarithm functions pass through the point (1, 0)
The six trigonometric functions of interest in our
Calculus class are: sine: f ( x ) = sin ( x )
cosine: f ( x ) = cos ( x )
tangent: f ( x ) = tan ( x )
cosecant: f ( x ) = csc ( x )
secant: f ( x ) = sec ( x )
cotangent: f ( x ) = cot ( x )
We focus on sine and cosine.
f(t) = sin(t) and g(t) = cos(t) are defined in terms of the
arc length t (measured in radians), and the
corresponding point on the unit circle.
http://cerebro.cs.xu.edu/~staat/Handouts/UnitCircle.pdf
Plotting these “special angles”, we get the following graphs:
f ( x ) = sin ( x )
f ( x ) = cos ( x )
Domain of sine and cosine is (-∞, ∞).
The range of sine and cosine is [-1, 1].
The other functions are all defined in terms of sine and
cosine, so knowing these two well allows us to work
with any of the others.
tan ( x ) =
csc ( x ) =
sin ( x )
cos ( x )
1
sin ( x )
1
sec ( x ) =
cos ( x )
cos ( x )
1
cot ( x ) =
=
tan ( x ) sin ( x )
The domain of each of these is determined
by the fact that denominators cannot be 0
and the following facts:
1. sinq = 0 Þ q = ±p n
2. cosq = 0 Þ q = p2 ± p n