Download INTRODUCTORY MATHEMATICAL ANALYSIS

2.1
Functions
Definition of a Function
A function is a correspondence between two sets of elements, Domain and
Range, such that to each element in the Domain there corresponds one and
only one element in the Range. If the correspondence is not a function, it
is a relation.
Example: Which specifies a function?
Domain
1
2
3
4
Range
5
7
Domain
2
3
Range
4
6
5
7
In a set of ordered pairs, (x,y), there is a correspondence between the first
numbers, x, and the second numbers, y. The set of containing x is the
Domain and the set containing y is the Range. This correspondence may or
may not be a function.
Example: Does the set of ordered pairs represent a function?
{(1,2),(2,5),(3,4),(4,4)}
Example: Does the set of ordered pairs represent a function?
{(1,2),(2,5),(3,4),(1,4)}
Functions Specified by Equations
y = 6x + 2
x is the independent variable or arguement
y is the dependent variable
If in an equation in two variables, we get exactly one value for the
dependent variable for each value of the independent variable, then the
equation defines a function. This can be determined by looking at the
explicit form of the equation.
Example: Determine if the equation specifies a function.
a)
3y - 6x = 3
b)
y2 - x4 = 9
Vertical-Line Test for a Function
An equation defines a function if each vertical line in the coordinate system
passes through at most one point on the graph of the equation.
Example: Use vertical-line test
Function Notation: y = f(x)
Given a function
y = 2x + 5
replacing the dependent variable y with the symbol f(x), “read f of x,”
gives
f(x) = 2x + 5
The Symbol f(x)
For any element x in the domain of f, the symbol f(x) represents the
element in the range of f corresponding to x in the domain of f. If x is an
input value, then f(x) is the corresponding output value. If x is an element
that is not in the domain of f, then f is not defined at x and f(x) does not
exist.
Example: Evaluating a function
If f(x) = 2x + 5, find the following:
a)
f(0)
b)
f(3)
c)
f(-2)
Example: For the function f(x) =
a)
f(5)
b)
f(-2)
c)
f(2)
5
, find
x2
Finding a Domain
The domain of a function is the set of all real numbers for which the
function is defined.
Example: Find the domain of each function.
a)
x2 – 3x + 1
b)
g(x) =
c)
h(x) =
6
x3
2x
Example: For f(x) = x2 – 4x + 9, find
a)
f(a)
b)
f(a + h)
c)
f(a  h)  f(a)
h
Example: Use the graph of the function f to answer the following
questions.
a)
Find f(0) and f(6).
b)
Find f(2) and f(-2).
c)
Is f(3) positive of negative?
d)
Is f(-1) positive or negative?
e)
For what numbers is f(x) = 0?
f)
For what numbers is f(x) < 0?
g)
What is th domain of f?
h)
What is the range of f?
i)
What are the x-intercepts?
j)
What is the y-intercept?
k)
How often does the line y = -1 intersect the graph?
l)
How often does the line x = 1 intersect the graph?
m)
For what value of x does f(x) = 3?
n)
For what value of x does f(x) = -2?