Chapter 4 : Rational function
A function f(x) is called a rational function if and only if it
can be written in the form
f(x) = \frac{P(x)}{Q(x)}
where P\, and Q\, are polynomials in x\, and Q\, is not the zero
polynomial. The domain of f\, is the set of all points x\, for
which the denominator Q(x)\, is not zero.
However, if \textstyle P and \textstyle Q have a non
constant polynomial greatest common divisor \textstyle R, then
setting \textstyle P=P_1R and \textstyle Q=Q_1R produces a
rational function f_1(x) = \frac{P_1(x)}{Q_1(x)},
which may have a larger domain than f(x), and is equal to f(x)
on the domain of f(x). It is a common usage to identify f(x) and
f_1(x), that is to extend "by continuity" the domain of f(x) to
that of f_1(x). Indeed, one can define a rational fraction as an
equivalence class of fractions of polynomials, where two
fractions A(x)/B(x) and C(x)/D(x) are considered equivalent if
A(x)D(x)=B(x)C(x). In this case \frac{P(x)}{Q(x)} is equivalent
to \frac{P_1(x)}{Q_1(x)}.
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Examples of rational functions
Other function
Piecewise – defined function
Piecewise-defined, or split definition, functions can be graphed on the
graphing calculator.
You will need to utilize the relational operators symbols
When using this method, enter each SECTION of the function into
a separate Y= area.
Graph
Parentheses needed!
The absolute value function
Graph y = | x + 2 |
One of the other students does what is commonly done: he picks only
positive x-values for his T-chart:
The real absolute value function is continuous everywhere. It is
differentiable everywhere except for x = 0. It is monotonically
decreasing on the interval (−∞,0] and monotonically increasing on
the interval [0,+∞). Since a real number and its opposite have the
same absolute value, it is an even function, and is hence not invertible.
Square – root function
he principal square root function f(x) = √ x (usually just referred to as
the "square root function") is a function that maps the set of nonnegative real numbers onto itself. In geometrical terms, the square root
function maps the area of a square to its side length.
The graph of the function f(x) = √ x, made up of half a parabola with a
vertical directrix.
Increasing and decreasing function
A function is "increasing" when the y-value increases as the xvalue increases, like this:
It
is easy to see that
y=f(x) tends to go up as it goes along.
Algebra of functions
The informal definition of an algebraic function provides a number of
clues about the properties of algebraic functions. To gain an intuitive
understanding, it may be helpful to regard algebraic functions as
functions which can be formed by the usual algebraic
operations: addition, multiplication, division, and taking an nth root.
Of course, this is something of an oversimplification; because of casus
irreducibilis (and more generally the fundamental theorem of Galois
theory), algebraic functions need not be expressible by radicals.
First, note that any polynomial function
is an algebraic
function, since it is simply the solution y to the equation
More generally, any rational function is
algebraic, being the solution to
Moreover, the nth root of any polynomial
algebraic function, solving the equation
is an
Surprisingly, the inverse function of an algebraic function is
an algebraic function. For supposing that y is a solution to
for each value of x, then x is also a solution of this
equation for each value of y. Indeed, interchanging the
roles of x and y and gathering terms,
One to one function and their inverses
Definitions:
• One-to-one function: is a function in which no two elements of the
domain A have the same image.
In other words, f is a one-to-one function if f(x1) = f(x2) implies x1 =
x2.
• Inverse function: Let f be a one-to-one function with domain A and
range B. Then its inverse
function, denoted f −1 has domain B and range A and is defined by f
−1
(y) = x if and only if f(x) = y , for any y in B
Chapter 5: Exponential and Logarithmic Functions
5.1 Exponential Functions and Their Graphs
is a function that can be expressed in the form f(x) = bx in which b>0 and b≠1. b is
called the base of exponential function.
Example:A
5.2 Logarithmic Functions and Their Graphs
The inverse of an exponential function is called a logarithmic function. For b>0,
b≠1,
Example:
The Natural Logarithmic Function
The Natural Logarithmic Function with base e is the natural logarithmic function
and is denoted by the specified symbol In x, read as “el en of x”
Common Logarithm
The logarithm with base 10 is called the common logarithm and is denoted by
omitting the base:
log10 x= log x
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a≠1 and b≠1. Then log a x is
given by
5.3 Properties of Logarithms
If b, M, and N are positive real numbers, b≠1 and k are real numbers.
5.4 Solving Exponential and Logarithmic Equations
Solving Expnential Equations
Equation with variables in the exponents, such as 2x =6, and 3-2x = 27, are called
exponential equations.
Exponential Property
If b>0 and b≠1, then bm =bn if and only if m=n in which m and n are any real
numbers.