1.2 Functions Defn. A function from a set D to a set R is a rule that

September 08, 2011
1.2
Functions
Defn. 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.
set D = all input (DOMAIN)
set R = all output (RANGE)
Function notation:
Ex.
1. f(x) = -x2+1
f(-1) =
y = f(x)
2. g(t) = -t2+4t + 1
g(x+2)=
September 08, 2011
Time of Day
8:00
9:00
10:00
11:00
Temp
55
60
63
65
70
Each x has exactly one y
(function)
Domain
1
2
3
4
5
Range
-5
-10
-15
-20
-25
An x has two y-values
(not a function)
Characteristics:
1. Each element in D must be matched with exactly one element of R.
2. Some elements of R may or may not be matched with any element of D
3. Two or more elements of D may be matched with the same element of R.
September 08, 2011
Which are functions?
1.
x 1 2 3 3
y 5 8 7 9
3. input = # state rep.
output = # state sen.
2.
x 1 2 3 4
y 5 5 8 9
4.
September 08, 2011
Equ.
y=x
2
(y depends on x so y is a function of x)
dep. indep.
domain: all values of x for which the function is defined
range: all values of y
Which is y a function of x?
1. x2+y = 1
2. -x + y2 = 1
September 08, 2011
Find the domain.
1. f(x) = 2x + 3
2. f(x) = 2
x+3
3. f(x) = 2 + 2
x
x+3
4. f(x) = √x
5. f(x) = √x+1
6. f(x) =
2
√4-x
x
September 08, 2011
Find the range.
1. f(x) = 2
x
2. f(x) = x
September 08, 2011
September 08, 2011
Continuity
continuous: does not come apart at any point.
continuous at all x
jump discontinuity
removable discontinuity
continuous everywhere
except at x = a
infinite discontinuity
removable discontinuity
f(a) doesn't exist
September 08, 2011
Which functions are discontinous at x = 2?
a.
b.
c.
September 08, 2011
Increasing/Decreasing Functions
increasing
decreasing
Example: Exploration 1 (Pg.93)
constant
decreasing (-∞, -1]
constant [-1, 1]
increasing [1, ∞)
September 08, 2011
Identify intervals on which function is increasing,
decreasing, or constant.
2
1. f(x)=(x+2) - 1
2
2. f(x) = x
2
x -1
September 08, 2011
Boundedness
unbounded
bounded below
bounded above
Identify the type of boundedness
1. f(x) = 3x2 - 4
2. f(x) = x
1 + x2
bounded
September 08, 2011
Local/Relative Extrema : extreme values
Identify local extrema in the graph:
f(x) = x4 - 7x2 + 6x
September 08, 2011
Symmetry: looks the same on both sides
(-x, f(x))
(x, f(x))
(x, f(x))
(-x, -f(x))
y = x2
y = x3
Even
Odd
Identify as even, odd, or neither.
1. f(x) = 3
1+x
2
2. f(x) = x3 + 0.04x2 + 3
3. f(x) = 1
x
September 08, 2011
Asymptotes: lines in which the graph is approaching but not
touching.
f(x) = 2x2
4 - x2
horizontal asymptotes:
vertical asymptotes:
x a- "x approaches a from the left "
x a+ "x approaches a from the right"
horiz:
lim f(x) = b
x -∞
vert: lim f(x) = ±∞
x alim f(x) = ±∞
x a+
September 08, 2011
Identify any asymptotes.
y=
x
2
x -x-2
September 08, 2011
End behavior.
What happens at the ends of the functions?
a. y = 3x
x2+1
b. y = 3x2
x2+1
c. y = 3x3
x2+1
d. y = 3x4
x2+1

Download Report