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Warm-up (10 min.)
I.


II.


Factor the following expressions completely over
the real numbers.
3x3 – 15x2 + 18x
x4 + x2 – 20
Solve algebraically and graphically
x(2x-1) = 10
_
x + x = 1
1.2 Functions and Their Properties
Today you will be able to:
Represent functions numerically,
algebraically, and graphically
Determine the domain and range for
functions
Analyze function characteristics such
as extreme values, symmetry,
asymptotes, and end behavior.
Parts of a Function
 Variables

These are the quantities in a function that can
change
 Dependent Variables

Depends on another quantity
 Independent Variables

These quantities tend to have an affect on the
dependent variable.
Parts of a Function - Example
 Variables
 Time and temperature
 Dependent Variables
 Temperature (depends on the time of day)
 Independent Variables
 Time (is unaffected by the temperature)
 Relating Variables

Often written as a ordered pair with the
independent variable first
(time, temperature)
Notation of a Function
 In general we use x to represent the
independent variable and y to represent the
dependent variable.
 We write…
y = f(x) meaning y is a function of x
T = f(t) meaning Temperature is a function of time
A Function Box
 We put in an input (independent variable)
apply a function and receive an output
(dependent variable).
FUNCTION
Input
Independent variable
Output
Dependent variable
f(x) = x + 2
4
6
Homework Check
 You have 5 min. to write down your answers
to the following homework questions

(No Talking!!)
 Exchange papers for grading.
 Give a score out of 10 and initial.
 Return to owner for reviewing and recording.
 Pass to your left (my right) for keeping.
 Questions on the homework?
Homework Answers
4) linear; H and O 8) quad; J and K
12) a – decreasing; b – 1960-1965
16) 64.7; 72.4
20) rectangle 24) a – calc; b – P=1.13x2+3.1x+443
c – about 2005 (x=18) d – not likely
28) A – players; b – owners; c - fans
32)
42)
45)
52)
b. (b/2)·(b/2) = (b/2)2
c. x2 + bx + (b/2)2 = (x + b/2)2 is the algebraic formula
completing the square, just as the area (b/2)2 completes
the
area (x + bx) to form the area (x + b/2)2.
Domain and Range
 Setting up our graph


While the coordinate plane continues infinitely
in all directions, most data will not.
Therefore we must determine a logical domain
for the independent data.


Domain: the set of logical values for the
independent variable
This will affect the range of values that makeup the dependent variable

Range: the set of values for the dependent
variable that correspond to specific independent
variables
Mappings
 Uses directed arrows to connect elements
of the domain to the elements of the range
5
10
15
20
0
3
5
Which mapping
represents a
function? Explain
5
0
10
3
15
5
20
Ex2 Does x = 2y2 define y as a function of x?
Vertical Line Test
A graph (set of points (x,y)) in the xy-plane defines y as
a function of x if and only if no vertical line intersects
the graph in more than one point.
What is your domain?
 Implied domain – the domain of a function
defined by an algebraic expression
 Relevant domain – the domain that fits a
given situation
Ex s2 has implied domain of all real numbers
However, if A = s2 is the area of a square then
the relevant domain is only non-negative real
numbers (s 0).
Ex 3 Pair with a neighbor to find the domain of each
function.
, where A(s) is the
area of an equilateral triangle with
side of length s.
Finding Range
Ex 4 Find the range of f(x) = 10 – x2.
Graphs of functions in reverse
Ex 5 Sketch the graph of a function that has
domain [-5,-1]  (2,4] and range [-4,-1] 
[1,).
Continuity
 Important property of the majority of functions
that model real-world behavior
 Graphically speaking, continuity investigates
at a point whether a function comes apart at
that point.
a
Continuous at all x = a
Discontinuous at x = a
Removable or Non-removable?
That is the question!!
 Discontinuities can be removable or non-
removable
Removable discontinuity
Removable discontinuity
a
Non-removable discontinuities
a
a
a
Infinite discontinuity
Jump discontinuity
Ex 6 Graph each of the following. Which of the
following are discontinuous at x = 2? Are any of the
discontinuities removable?
a) f(x) = x + 3
x–2
b) g(x) = (x + 3)(x – 2)
c) h(x) = x2 – 4
x–2
Ex 7 Label each graph (or parts) as increasing,
decreasing, or constant.
1
1
1
1
1
1
1
1
Increasing, Decreasing, and Constant
Function on an Interval
 A function f is increasing on an interval if, for any
two points in the interval, a positive change in x
results in a positive change in f(x).
 A function f is decreasing on an interval if, for any
two points in the interval, a positive change in x
results in a negative change in f(x).
 A function f is constant on an interval if, for any two
points in the interval, a positive change in x results in
a zero change in f(x).
Ex 8 Tell the intervals on which each function is
increasing and the intervals on which it is
decreasing.
f(x) = (x - 2)2
g(x) =
x2
x2 - 1
Boundedness
 A function f is bounded below if there is some
number b that is less than or equal to every number
in the range of f. Any such number b is called a
lower bound of f.
 A function f is bound above if there is some number
B that is greater than or equal to every number in
the range of f. Any such number B is called an
upper bound of f.
 A function f is bounded it is bounded both above and
below.
Ex 9 Identify each of these functions as bounded
below, bounded above, or bounded.
w(x) = -3x2 + 4
p(x) =
x
x2+1
Warm-Up (8 min.)
Determine the domain and range of each
function.
1. y = 1+ x
2. f(x) = x2 - 16
x
State the intervals on which each graph above
is increasing or decreasing.
Local and Absolute Extrema
 A local maximum of a function f is a value f(x) that is
greater than or equal to all range values of f on some
open interval containing c. If f(x) is greater than or
equal to all range values of f, then f(c) is the
maximum (or absolute maximum) value of f.
 A local minimum of a function f is a value f(x) that is
less than or equal to all range values of f on some
open interval containing c. If f(x) is less than or
equal to all range values of f, then f(c) is the minimum
(or absolute minimum) value of f.
 Local extrema are also called relative extrema.
Ex 10 Decide whether f(x) = x3 – 2x2 + 5x – 2 has
any local extrema.
Symmetry
 A graph appears identical when viewed on
either side of a line or through a point.
 Symmetry can be easily viewed graphically,
numerically, or algebraically.
Summarizing symmetry
Type of
Symmetry
Graphically
With respect to
the y-axis
f(x) = x2
With respect to
the x-axis
x = y2
With respect to
the origin
f(x) = x3
Numerically
Algebraically
f(-x) = f(x)
called even
functions
*Not a function
but (x,-y) is on
the graph
whenever (x,y)
is on the graph
f(-x) = - f(x)
called odd
functions
Ex 11 Tell whether each of the following functions
is even, odd, or neither.
f(x) = x2 – 3
b) g(x) = x2 – 2x – 2
c) h(x) =
x3
4 – x2
a)
Asymptotes
 The line y = b is a horizontal asymptote of the
graph of a function y = f(x) if f(x) approaches a
limit of b as x approaches + ∞ or - ∞.
 In limit notation,
 The line x = a is a vertical asymptote of the
graph of a function y = f(x) is f(x) approaches a
limit of + ∞ or
-∞ as x approaches a from either
direction.
 In limit notation,
Ex 12 Identify any vertical or horizontal asymptotes by
viewing each graph and observing its end behavior.
Tonight’s Assignment
 P. 98-100
Ex 3-66 m. of 3