Do Now from 1.1b…
Solve the equation graphically by converting it into
an equivalent equation with 0 on the right-hand
side and then finding the x-intercepts
3x  2  2 ( x  8)
x  1.09, 2.86
( x  6)  6  2 (5  x)
x  2.66
Functions and their
properties…domain, range,
continuity, discontinuity.
We are functioning well
in Sec. 1.2a!!!
Definition: Function, Domain, and Range
A function from a set D to a set R is a rule that
assigns to every element in D a unique element in R.
The set D of all input values is the domain of the
function, and the set R of output values is the range
of the function.
Common notation: y = f(x)
Here, x is the independent variable,
and y is the dependent variable
VLT!!! (not a tasty BLT sandwich…)
Vertical Line Test: A graph (set of points (x, y)) in
the x-y plane defines y as a function of x if and only
if no vertical line intersects the graph in more than
one point.
Which of the following are graphs of functions?
Yup!!!
Yessir!!!
Nope!!!
Heck Naw!!!
Finding Domain and Range
Agreement for Domain: Unless we are dealing with
a model (like volume) that necessitates a restricted
domain, we will assume that the domain of a function
defined by an algebraic expression is the same as the
domain of the algebraic expression, the implied
domain. For models, we will use a domain that fits
the situation, the relevant domain.
Finding Domain and Range
Find the domain of the following functions (support graphically):
f  x  x  3
The key question: Is there anything that x could not be???
x 3 0
x  3
Always write your answer
in interval notation:
D :  3,  
Finding Domain and Range
Find the domain of the following functions (support graphically):
x
g  x 
x 5
What are the restrictions on x ?
x 5  0
x5
x0
Interval notation:
D : 0,5
5,  
Finding Domain and Range
Find the range of the given function (use any method).
g  x  5  4  x
What are the possible y-values for this function???
R : 5, 
Finding Domain and Range
Find the range of the given function (use any method).
3 x
g  x 
2
4 x
2
Check the graph…
R :  , 1
3 4, 
The Concept of Continuity
Algebraically, a function is continuous at x = a if
lim f  x   f  a 
x a
Read “the limit of f (x) as x approaches a is f (a)”
Graphically, a function is continuous at a particular point if the
graph does not “come apart” at that point.
 Let’s apply this with some examples…
The Concept of Continuity
Continuous at all x
How does that
“limit definition”
apply???
The Concept of Continuity
Removable Discontinuity
at x = a
a
Why is it called
“removable”?
The Concept of Continuity
Jump Discontinuity
at x = a
a
The Concept of Continuity
Infinite Discontinuity
at x = a
a
Homework: p. 98 1-23 odd