Theme: Definitions and Representations
Topic: Functions and Limits of Sequences
Today
7:10-∼8pm – Functions (they are everywhere!)
∼8pm-8:30pm – Sequences and Limits
Definition of Function
Warm-up: Among the following, which best defines a function?
A function is . . .
(a) . . . a rule that takes inputs and assigns to each input exactly one number as output.
(b) . . . a formula that takes numbers as inputs and assigns to each input exactly one
number as output.
(c) . . . a rule that takes inputs and assigns inputs to outputs. The inputs and outputs
are numbers.
(d) . . . a rule that takes inputs and assigns each input to exactly one output, and where
each output is associated to exactly one input.
Please wrap up your thoughts on this by 7:20pm.
Warm Up – please wrap up by 7:20pm
Among the following, which best defines a function?
A function is . . .
(a) . . . a rule that takes inputs and assigns to each input exactly one number as output.
(b) . . . a formula that takes any real number as an input and assigns to each input
exactly one number as output.
(c) . . . a formula that takes inputs and assigns inputs to outputs. The inputs and
outputs are numbers.
(d) . . . a rule that takes inputs and assigns each input to exactly one output, and where
each output is associated to exactly one input.
.
. .
.
.
census count
of MI population
(To substantiate your choice, consider which of the examples I-V below
fit/are ruled out by (a)-(d).)
IV
I
III
V
II
..
.. . .
years since 1950
Using the Definition of Function: Summary
A function is a rule that takes inputs and assigns to each input exactly
one number as output.
Four representations of functions prevalent in US textbooks:
symbolic, graphical, tabular, verbal.
To use a representation, you must know how to describe the components
of the definition of a function:
domain (inputs),
range (where outputs can live)*,
rule from inputs to outputs,
in terms of that representation.
(*we will use image to refer to the set of actual outputs.)
Using notation/symbolic representation
In Algebra I homework in September, a student has written the following.
Which
calculations are mathematically reasonable? What’s the reasoning?
a
5
(a) 7 = 7
/7a = /75
a=5
(b) log 3a = log 15
−
log
−
− 3a =−
log
−
− 15
3a = 15 → a = 5
(c) (5a)2 = 252
(5a)/2 = 25/2
5a = 25 → a = 5
√
√
25
(d) √5a = √
\5a = \25
5a = 25 → a = 5
Using notation/symbolic representation
In Algebra I homework in September, a student has written the following.
Which
calculations are mathematically
reasonable? What’s the reasoning?
x
a
5
(a) 7 = 7
/7a = /75
a=5
(a) Let f (x) = 7 .
f (a) = f (5)
\f (a) = \f (5) → a = 5
(b) log 3a = log 15 (b) Let f (x) = log(x).
f (3a) = f (15)
−
log
−
− 3a =−
log
−
− 15
f\(3a) = f\(15) → a = 5
3a = 15 → a = 5
(c)
(c) (5a)2 = 252
(5a)/2 = 25/2
5a = 25 → a = 5
√
√
(d)
25
(d) √5a = √
\5a = \25
5a = 25 → a = 5
Let f (x) = x 2 .
f (5a) = f (25)
f\(5a) = f\(25) → a = 5
√
Let f (x) = x.
f (5a) = f (25)
f\(5a) = f\(25) → a = 5
Using notation/symbolic representation
In Algebra I homework in September, a student has written the following.
Which
calculations are mathematically
reasonable? What’s
the reasoning?
x
x
a
5
(a) 7 = 7
/7a = /75
a=5
(a) Let f (x) = 7 .
f (a) = f (5)
\f (a) = \f (5) → a = 5
(a) Let f (x) = 7 .
f (a) = f (5)
f −1 (f (a)) = f −1 (f (5)) → a = 5
(b) Let f (x) = log(x).
(b) log 3a = log 15 (b) Let f (x) = log(x).
f (3a) = f (15)
f (3a) = f (15)
−
log
−
− 3a =−
log
−
− 15
f\(3a) = f\(15) → a = 5
f −1 (f (3a)) = f −1 (f (15)) → a = 5
3a = 15 → a = 5
(c)
(c) (5a)2 = 252
(5a)/2 = 25/2
5a = 25 → a = 5
√
√
(d)
25
(d) √5a = √
\5a = \25
5a = 25 → a = 5
Let f (x) = x 2 .
(c) Let f (x) = x 2 .
f (5a) = f (25)
f (5a) = f (25)
f\(5a) = f\(25) → a = 5
f −1 (f (5a)) = f −1 (f (25)) → a = 5
√
√
Let f (x) = x.
(d) Let f (x) = x.
f (5a) = f (25)
f (5a) = f (25)
f\(5a) = f\(25) → a = 5
f −1 (f (5a)) = f −1 (f (25)) → a = 5
Inverse Functions
A common characterization of inverse function is:
the inverse function “undoes” the original function
When is there an inverse function? What does it mean to “undo”?
Inverse Functions
A common characterization of inverse function is:
the inverse function “undoes” the original function
When is there an inverse function? What does it mean to “undo”?
If and only if the function is injective (one-to-one).
An injective function a rule that takes inputs and assigns each input to
exactly one output, and where each output is associated to exactly one
input.
[graphical explanation on blackboard]
Left vs. right composition of inverses
What do you think of the following reasoning?
√
x2 =
√
\ x/2 =
x
p
q
(−5)
\ /2
because
f
=
√
x is invertible
f −1 (x)
=
x2
(f (x))
√
x2
=
=
x (“undoing”)
√
x is like \ x/2
f (x)
=
√
x is invertible
f −1 (x)
=
x2
=
x (“undoing”)
√
x is like ( \x)/2
−1
−5
=
√
( x)2 =
√
( \x)/2 =
x
(−5)2
f (x)
=
√
( 5)2
√
( \5)/2
5
because
f
−1
(f (x))
√
( x)2
=
This Week’s Work on Functions:
Summary/Preview
Definition as way to organize/think about representing functions
Prevalent representations of functions in US textbooks
Invertibility ⇔ one-to-one (they are equivalent)
f −1 ◦ f vs. f ◦ f −1 have different domains.
When f invertible, usually begin by defining f −1 as function such that
f −1 (f (x)) = x.
(Next time:) Making sense of inverting sine, cosine!
Though the above convention is used for arcsin and arccos (which are
the invertible functions), the inverse notation for arcsin and arccos can
be confusing.
This Week’s Work on Functions:
Summary/Preview
Definition as way to organize/think about representing functions
Prevalent representations of functions in US textbooks
Invertibility ⇔ one-to-one (they are equivalent)
f −1 ◦ f vs. f ◦ f −1 have different domains.
When f invertible, usually begin by defining f −1 as function such that
f −1 (f (x)) = x.
(Next time:) Making sense of inverting sine, cosine!
Though the above convention is used for arcsin and arccos (which are
the invertible functions), the inverse notation for arcsin and arccos can
be confusing.
And now . . .
Limits and Sequences.
Sequences, Limits, and Convergence
Definition. A sequence (of real numbers) is a function from N to R.
Sequences, Limits, and Convergence
Definition. A sequence (of real numbers) is a function from N to R.
In your notebooks, silently work on the following problem:
Come up with a definition of when a sequence converges.
Sequences, Limits, and Convergence
Definition. A sequence (of real numbers) is a function from N to R.
In your notebooks, silently work on the following problem:
Come up with a definition of when a sequence converges.
In groups:
Discuss and revise your definitions of convergence and limit. Use
the following sequences to examine your definition.
an =
1
n
an =
an =
0, if n is odd
1
n , if n is even
√
an = 1
n an =
0
n
an = (−1)n +
1
n
An Armamentarium of Sequences
Represent the sequences below numerically, using input/output tables.
What patterns do you observe in the numerical values?
How do the terms change as the index gets larger?
an =
1
n
1,
1
1
−
1 n,
n , if
an =
1
10 , if
an =
an =
if n is odd
if n is even
n ≤ 10
n > 10
√
n
an =
n
5n
an = 1
an = (−1)n +
an = (−1)n (1 + n1 )
1, if n is odd
an =
1
if n is even
n
an = n3 /2n
1
n
Representing sequences graphically
Which sequences are these the graphs of?
Is there anything surprising about these graphs?
What might a high school student find surprising about these graphs?
This Week’s Work on Sequences
Sequences are functions with N as their domain
Convergence
vs. cluster point/accumulation point
vs. approaching but not touching
Convergence/divergence in graphical and tabular representations
(Next time:) . . . -strips – a way of gaining intuition for the formal
definition of convergence!