Transcript PRECALCULUS I Functions and Graphs •Function, domain, independent variable •Graph, increasing/decreasing, even/odd Dr. Claude S.

PRECALCULUS I
Functions and Graphs
•Function, domain, independent variable
•Graph, increasing/decreasing, even/odd
Dr. Claude S. Moore
Danville Community College
1
Definition: Function
A function f from set A to set B is a rule
of correspondence that assigns to each
element x in set A exactly one
element y in set B.
Set A is the domain (or set of inputs) of
the function f, and set B contains
range (or set of outputs).
Characteristics of a Function
1. Each element in A (domain) must be
matched with an element of B (range).
2. Each element in A is matched to not more
than one element in B.
3. Some elements in B may not be matched
with any element in A.
4. Two or more elements of A may be
matched with the same element of B.
Functional Notation
Read f(x) = 3x - 4 as “f of x equals three
times x subtract 4.”
x inside parenthesis is the
independent variable.
f outside parenthesis is the
dependent variable.
For the function f(x) = 3x - 4,
f(5) = 3(5) - 4 = 15 - 4 = 11, and
f(-2) = 3(-2) - 4 = - 6 - 4 = -10.
Piece-Wise Defined Function
A “piecewise function” defines the function
in pieces (or parts).
In the function below,
if x is less than or equal to zero,
f(x) = 2x - 1; otherwise, f(x) = x2 - 1.
 2 x  1 if x  0
f ( x)   2
 x  1 if x  0
Definition: Function
A function f from set A to set B is a rule
of correspondence that assigns to each
element x in set A exactly one
element y in set B.
Set A is the domain (or set of inputs) of
the function f, and set B contains
range (or set of outputs).
Piece-Wise Defined Function
A “piecewise function” defines the function
in pieces (or parts).
In the function below,
if x is less than or equal to zero,
f(x) = 2x - 1; otherwise, f(x) = x2 - 1.
 2 x  1 if x  0
f ( x)   2
 x  1 if x  0
Domain of a Function
Generally, the domain is implied to be the set
of all real numbers that yield a real
number functional value (in the range).
Some restrictions to domain:
1. Denominator cannot equal zero (0).
2. Radicand must be greater than or equal to
zero (0).
3. Practical problems may limit domain.
Domain of a Function
Generally, the domain is implied to be the set
of all real numbers that yield a real
number functional value (in the range).
Some restrictions to domain:
1. Denominator cannot equal zero (0).
2. Radicand must be greater than or equal to
zero (0).
3. Practical problems may limit domain.
Summary of Functional Notation
In addition to working problems, you should
know and understand the definitions of
these words and phrases:
dependent variable
independent
variable
domain
range
function
functional notation
functional value
implied domain
Vertical Line Test for a Function
A set of points in a coordinate
plane is the graph of
y as a function of x
if and only if no vertical line
intersects the graph at more than
one point.
Increasing, Decreasing, and
Constant Function
On the interval containing x1 < x2,
1. f(x) is increasing if f(x1) < f(x2).
Graph of f(x) goes up to the right.
2. f(x) is decreasing if f(x1) > f(x2).
Graph of f(x) goes down to the right.
On any interval,
3. f(x) is constant if f(x1) = f(x2).
Graph of f(x) is horizontal.
Even and Odd Functions
1. A function given by y = f(x) is even if,
for each x in the domain,
f(-x) = f(x).
2. A function given by y = f(x) is odd if,
for each x in the domain,
f(-x) = - f(x).