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CHAPTER 1
A function f is a rule that assigns to
each element x in a set A exactly one
element, called f (x), in a set B.
The set A is called the domain of
the function.
The range of the function is the set
of all possible values of f (x) as x
varies throughout the domain.
Increasing and Decreasing
Functions
A function f f is called increasing
on an interval I if
f(x1) < f(x2)
whenever x1 < x2 in I.
Increasing and Decreasing
Functions
A function f f is called decreasing
on an interval I if
f(x1) > f(x2)
whenever x1 < x2 in I.
1.5 Exponential Functions
An exponential function is a
function is a function of the form
f(x) = ax where a is a positive
constant.
If x = n, a is a positive integer,
then an = a .a . a… . … . a (n factors)
If x = 0,then a0=1, and if x = -n
where n is a positive integer,
then a-n=1/an .
If x is a rational number, x= p/q
where p and q are integers and
q > 0, then ax= a p/q = qa p .
Laws of Exponents If a and b
are positive numbers and x and
y are any real numbers, then
1. ax+y = ax ay
2. ax-y = ax /ay
3. (ax) y = ax y
4. (ab)x = axby
Definition of E In the family of
exponential functions f(x) = bx
there is exactly one exponential
Function for which the slope of the
Line tangent at (0,1) is exaclty 1.
This occurs for b=2.71…This
Important number is denoted by e.
1.6 Inverse Functions and
Logarithms
Definition A function f is called a
one-to-one function if it never
takes on the same value twice; that
is,
f(x ) is not equal to f(x2)
whenever x1 is not equal to x2 .
1
Horizontal Line Test A function is
one-to-one if and only if no
horizontal line intersects its graph
more than once.
1.6 Inverse Functions and
Logarithms
Definition Let f be a
one-to-one function with domain
A and range B. Then its inverse
function f –1 has domain B and
range A and is defined by
f –1(y) = x, then f(x) = y
for any y in B.
domain of f –1 = range of f
range of f –1 = domain of f .
f –1 (x) = y then f(y) = x.
f –1 (f (x)) = x for every x in A
f (f –1(x)) = x for every x in B.
The graph of f –1 is obtained by
reflecting the graph of f about the
line y = x.
The graph of f –1 is obtained by
reflecting the graph of f about the
line y = x.
ln x = y then e y = x
The graph of f –1 is obtained by
reflecting the graph of f about the
line y = x.
Laws of Logarithms
If x and y are positive numbers,
then
• ln (x y) = ln x + ln y
• ln (x/y) = ln x - ln y
• ln (xr) = r ln x (where r is
any real number)
ln x = y then e y = x
ln x = y then e y = x
ln(e x) = x
e ln x = x
xR
x>0
ln x = y then e y = x
ln(e x) = x
e ln x = x
ln e = 1
xR
x>0
For any positive number a ( a
is not equal to 1), we have
log a x = ln x / ln a