Theme 1 -
unit 1.
↳
describe a function
>
estimate function values and intersections
>
lise the graph of
a function
Describing a function Definition of a function
& Stuff in (input) * function is a rule which takes numbers
4)
as inputs and assigns to each input exactly
functione One output
- Stuff out (output) ↳ more than one output = not a function
One
thing in
gives one thing out
V
functions can't
restrictions
work with all values and have
& inconscion ⑧
↳
eg -x =
0 kills the function one output per input
The input is the independent variable Idoesn't need unique outputs)
: the output is the dependent variable
Always work with real numbers (R) to get real outputs from real inputs
function
↓ isinput label
function notation
Ab
label 3
as A(i) in
C =
↳ doutput label outd &
function
xY M
>
- function
The vertical line test X
I
X & function
Shows if a graph isa X
-
X
function
S
X
> x <X S
~
not ad ↳
More than one V
function
crossing at a point = not a function
The collection of inputs of a function is the domain of a function describes the
function/
· the collection
f
of outputs of a function is the
range of a function function tells US what
it does &
3 A =+(f)
L
if 1259 e
"
ep .
: it seems reasonable to conclude that A =
=
2
A 14258 output e &
in put
↓ belongs
to the
sety
3 3
[3
Set notation
R both have exactly
:
anything inside nat means
↑ &
the
condition to
same meaning
indicates a set to be in "
"such that be in the set
set
ep .
31 ,
2
,
33 the
read as real
describes the set : the get containing a such a
that is a
containing 1 2
,
and 3 number
,> additional conditions can be added by comma between them
using a
eg .
Exlece ,
O 3
the such that number between 0 and I,
set
containing is real
·
a I a
including 0 and excluding 1 (meaning
>
abuse of formal notation doesn't introduce o as the subject but means the same
Esler 3 3 Sceloxx 3
We will always
,
O > I
work with real numbers
formal notation abuse of formal notation so can drop E
Ex 1013 or Soker 0x13 ,
or
Soceloxe
All of these notations mean exactly the same
thing h
Interval notation
set notation :
Secloce ,
axx b3
take two
where a
real
is less
numbers
than
"Quick" notation :
(cc1D_x b 3 >
b (a<b) fix/ number
[C by
and smaller has to be
Interval notation :
,
get the numbers written first
↳
When a or b are included in the set then a square bracket] is used
~
when a or b are excluded in the set then a round bracket) is used
can find of real
you numbers numbers-
·
an infinite amount between two > a
"
continuous" set of numbers
·
interval straight line
ray
· ·
·
interval types : (closed interval] ,
Copen interval) , [half open) ,
(half closed]
·
can also show if a point on a function is included or excluded
↳ o =
excluded
g ⑧
= included I I >R
Note :
* the arrow on the end of a line shows which direction numbers are
increasing. Only use >
-
on a Cartesian plane or number line and NOT the
infinite lines on functions (just use a line without a dot on the end
a b C d
·
the Overlap is called the intersection I I I I >R
⑧ ⑨
O 8
↳ means "and"-what the Sets have in common g ⑨
> AlB = A and B at the same time intersection : [a c]n (b d)
,
.
= (b , C]
union : [a c]u(b d) [a d)
"
·
a union means " or , ,
=
,
↳ and
puts sets intervals together
~ AUB =
either A or B
Domain : all allowed values as inputs of a function
↳ written as : dom (f) ,
domf or 7(f)
ep dom (f)
.
=
SolecER ,
f() exists
,Range : all output numbers of a function
have to include the restriction
&
range (f)
↳ written as : or R(f)
ep range (f)
.
=
EylyER , y
= f(x) for some oce dom (f)3
NB ! Graphs are always read from left to
right to determine increasing or
decreasing
&gincreasing
↳ a function f , ,
is increasing at an interval if
decreasing 1
the f(x) value is increasing as the s value
is increasing
~ a function I , ,
is
decreasing at an interval if
the f(x) value is decreasing as the x value
is
increasing
↑ fled = i in the
figure is always decreasing
GoIE
domce) 03
3
= but e *
> get notation
ExxxeR 03
7
x =
=
,
=
(-0 , 0)U(0 ,
0) Sinterval notation
Concavity : a function f , ,
is concave up at an interval if the graph of f lies
above all of its
tangent lines on the interval
V
a
concave
up
parabola exponential
↳ from
concavity can
change concave up to concave down and visa-versa
~
concave down a function f is down interval if f
concave at the
graph of
:
, ,
an
lies below all of its tangent lines on the interval
&
Concave down y
M -
parabola
logarithm
Unit 1 .
2
Concept : some numbers are fixed and have
symbols associated
↳ Eller's number is denoted as "e"
natural log the oe
loges == In-
the
·
e- >
,
-
When talking about the exponential function= e
-
when talking about the logarithm
=
Inc
Some functions to know
1..) linear functions
↳
y
mx +
cary = ax + b
=
·
D. DER are constants in a
given equation
domain (ax + b) R
·
=
, range of deta is dependent
·
on a
=
when a =
0.:
y
= b.:
range = Eb3
When a to then R if using value e mac, C
-
range y
+
= =
,
M mx + C
x= k + N -m = 0 -
y =
: (x mx + d
E < &m ,
m = undefined y = C
o
is positive
C
> >
·
different form
↓
m is
negative just a :
-y = ax + b -
a straight line is just x' + C
>
a linear function : (constant) (independent variable)' + constant
3
>
the best description of a linear function is Ax +
By =
Ax and
By Can't both be O time important for
·
at the same
0 0 is & A and B the linear
tautology are constants algebra
·
=
·
0 =
5 is contradictory
↳ A = 0
,
BE0 :
By = C :
y =
= (horizontal line)
C
AO B 0 : Ax C :
(vertical line)
>
y a
= = =
,
- At0
,
B0 : Ax +
By =
y
= -x +
= (increasing or decreasing
2) exponential functions linear function (
↳ q b
y or
y
= =
b
#increas >
=
-
-
a
decreasing
S
p
-
= Te
-
- I
⑧
y
=
p
-
-y
- -
= 0
<
prefer to have a whole number and negative exponent for
decreasing functions
>
important exponential function f(x) e
y
: = =
-
domain (f) = D
-range (f) (0 x) =
,
.
3)
) logarithmic function
↳
y
=
logb(x) or
y
=
loga(x)
opp-inverse
of increasing exponential functione
function
y
=
loga(x) inverse of decreasing exponential
-
=
>
10pp()
-
lopal = 0 and logay is undefined when
y
3
-
domain(f) = (0 ) inverse of exponential
,
-
range (f) = R
<
the important logarithm is the natural logorithm : In
4. ) functions
power
↳
y
= ax ,
deSPICE ,
a+
03
>
Special cases : a= 1 ,
neX
-
y = ax" = x
unit 1.
↳
describe a function
>
estimate function values and intersections
>
lise the graph of
a function
Describing a function Definition of a function
& Stuff in (input) * function is a rule which takes numbers
4)
as inputs and assigns to each input exactly
functione One output
- Stuff out (output) ↳ more than one output = not a function
One
thing in
gives one thing out
V
functions can't
restrictions
work with all values and have
& inconscion ⑧
↳
eg -x =
0 kills the function one output per input
The input is the independent variable Idoesn't need unique outputs)
: the output is the dependent variable
Always work with real numbers (R) to get real outputs from real inputs
function
↓ isinput label
function notation
Ab
label 3
as A(i) in
C =
↳ doutput label outd &
function
xY M
>
- function
The vertical line test X
I
X & function
Shows if a graph isa X
-
X
function
S
X
> x <X S
~
not ad ↳
More than one V
function
crossing at a point = not a function
The collection of inputs of a function is the domain of a function describes the
function/
· the collection
f
of outputs of a function is the
range of a function function tells US what
it does &
3 A =+(f)
L
if 1259 e
"
ep .
: it seems reasonable to conclude that A =
=
2
A 14258 output e &
in put
↓ belongs
to the
sety
3 3
[3
Set notation
R both have exactly
:
anything inside nat means
↑ &
the
condition to
same meaning
indicates a set to be in "
"such that be in the set
set
ep .
31 ,
2
,
33 the
read as real
describes the set : the get containing a such a
that is a
containing 1 2
,
and 3 number
,> additional conditions can be added by comma between them
using a
eg .
Exlece ,
O 3
the such that number between 0 and I,
set
containing is real
·
a I a
including 0 and excluding 1 (meaning
>
abuse of formal notation doesn't introduce o as the subject but means the same
Esler 3 3 Sceloxx 3
We will always
,
O > I
work with real numbers
formal notation abuse of formal notation so can drop E
Ex 1013 or Soker 0x13 ,
or
Soceloxe
All of these notations mean exactly the same
thing h
Interval notation
set notation :
Secloce ,
axx b3
take two
where a
real
is less
numbers
than
"Quick" notation :
(cc1D_x b 3 >
b (a<b) fix/ number
[C by
and smaller has to be
Interval notation :
,
get the numbers written first
↳
When a or b are included in the set then a square bracket] is used
~
when a or b are excluded in the set then a round bracket) is used
can find of real
you numbers numbers-
·
an infinite amount between two > a
"
continuous" set of numbers
·
interval straight line
ray
· ·
·
interval types : (closed interval] ,
Copen interval) , [half open) ,
(half closed]
·
can also show if a point on a function is included or excluded
↳ o =
excluded
g ⑧
= included I I >R
Note :
* the arrow on the end of a line shows which direction numbers are
increasing. Only use >
-
on a Cartesian plane or number line and NOT the
infinite lines on functions (just use a line without a dot on the end
a b C d
·
the Overlap is called the intersection I I I I >R
⑧ ⑨
O 8
↳ means "and"-what the Sets have in common g ⑨
> AlB = A and B at the same time intersection : [a c]n (b d)
,
.
= (b , C]
union : [a c]u(b d) [a d)
"
·
a union means " or , ,
=
,
↳ and
puts sets intervals together
~ AUB =
either A or B
Domain : all allowed values as inputs of a function
↳ written as : dom (f) ,
domf or 7(f)
ep dom (f)
.
=
SolecER ,
f() exists
,Range : all output numbers of a function
have to include the restriction
&
range (f)
↳ written as : or R(f)
ep range (f)
.
=
EylyER , y
= f(x) for some oce dom (f)3
NB ! Graphs are always read from left to
right to determine increasing or
decreasing
&gincreasing
↳ a function f , ,
is increasing at an interval if
decreasing 1
the f(x) value is increasing as the s value
is increasing
~ a function I , ,
is
decreasing at an interval if
the f(x) value is decreasing as the x value
is
increasing
↑ fled = i in the
figure is always decreasing
GoIE
domce) 03
3
= but e *
> get notation
ExxxeR 03
7
x =
=
,
=
(-0 , 0)U(0 ,
0) Sinterval notation
Concavity : a function f , ,
is concave up at an interval if the graph of f lies
above all of its
tangent lines on the interval
V
a
concave
up
parabola exponential
↳ from
concavity can
change concave up to concave down and visa-versa
~
concave down a function f is down interval if f
concave at the
graph of
:
, ,
an
lies below all of its tangent lines on the interval
&
Concave down y
M -
parabola
logarithm
Unit 1 .
2
Concept : some numbers are fixed and have
symbols associated
↳ Eller's number is denoted as "e"
natural log the oe
loges == In-
the
·
e- >
,
-
When talking about the exponential function= e
-
when talking about the logarithm
=
Inc
Some functions to know
1..) linear functions
↳
y
mx +
cary = ax + b
=
·
D. DER are constants in a
given equation
domain (ax + b) R
·
=
, range of deta is dependent
·
on a
=
when a =
0.:
y
= b.:
range = Eb3
When a to then R if using value e mac, C
-
range y
+
= =
,
M mx + C
x= k + N -m = 0 -
y =
: (x mx + d
E < &m ,
m = undefined y = C
o
is positive
C
> >
·
different form
↓
m is
negative just a :
-y = ax + b -
a straight line is just x' + C
>
a linear function : (constant) (independent variable)' + constant
3
>
the best description of a linear function is Ax +
By =
Ax and
By Can't both be O time important for
·
at the same
0 0 is & A and B the linear
tautology are constants algebra
·
=
·
0 =
5 is contradictory
↳ A = 0
,
BE0 :
By = C :
y =
= (horizontal line)
C
AO B 0 : Ax C :
(vertical line)
>
y a
= = =
,
- At0
,
B0 : Ax +
By =
y
= -x +
= (increasing or decreasing
2) exponential functions linear function (
↳ q b
y or
y
= =
b
#increas >
=
-
-
a
decreasing
S
p
-
= Te
-
- I
⑧
y
=
p
-
-y
- -
= 0
<
prefer to have a whole number and negative exponent for
decreasing functions
>
important exponential function f(x) e
y
: = =
-
domain (f) = D
-range (f) (0 x) =
,
.
3)
) logarithmic function
↳
y
=
logb(x) or
y
=
loga(x)
opp-inverse
of increasing exponential functione
function
y
=
loga(x) inverse of decreasing exponential
-
=
>
10pp()
-
lopal = 0 and logay is undefined when
y
3
-
domain(f) = (0 ) inverse of exponential
,
-
range (f) = R
<
the important logarithm is the natural logorithm : In
4. ) functions
power
↳
y
= ax ,
deSPICE ,
a+
03
>
Special cases : a= 1 ,
neX
-
y = ax" = x
