English | Español

Try our Free Online Math Solver!

Online Math Solver

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Function Sequence and Summation

Function

• Suppose A and B are nonempty sets. A function from A to B is an assignment of exactly one
element of B to each element of A.

 – We write f: A→B.
 – We write f(a)=b if b is the element of B assigned to element a of A.

• Example:

f: Z → Z
where for each x ∈ Z, f(x)=x^2.

Domain, codomain, range

• Suppose f is a function from A to B.
 – We say A is the domain of f.
 – We say B is the codomain of f.
 – We say {f(x) | x ∈ A } is the range of f.

Example:

f: Z → Z
where for each x ∈ Z, f(x)=x^2.
• Domain of f: Z
• Codomain of f: Z
• Range of f: {x|x=y^2, y ∈ Z}

Image and preimage

• Suppose f is a function from A to B and f(x)=y.
 – We say y is the image of x.
 – We say x is an preimage of y.
 – Note that the image of x is unique but there can be more than one preimages for y.

• Example:

f: Z → Z
where for each x ∈ Z, f(x)=x^2.
 • Image of 2: 4
 • Preimage of 4: 2
 • Another preimage of 4: -2

• Note that every element in the domain has an image.
 – But not every element in the codomain has a preimage. Only those in the range have preimages.

• Example:
f: Z → Z
where for each x ∈ Z, f(x)=x^2.
 • Each x ∈ Z has an image f(x)=x^2 .
 • But negative integers in Z do not have preimages.
 • Only perfect squares (i.e., those in the range) have preimages.

One-to-one function

• A function is one-to-one if each element in
the range has a unique preimage.
– Formally, f: A→B is one-to-one if f(x)=f(y)
implies x=y for all x ∈A,y ∈A.
• Example:

f: Z → Z
where for each x ∈ Z, f(x)=x^2.
• Clearly, f is NOT one-to-one because 4 has two preimages

Example one-to-one function

• f: N → Z
where for each x ∈ N, f(x)=x+5.
• f: Z+ → Z+
where for each x ∈ Z+, f(x)=x^2.
• f:{0,1,2} →{0,1,2,3}
where f(0)=1, f(1)=3, f(2)=2.

Onto function

• A function is onto if each element in
codomain has an image (i.e., codomain = range).
 – Formally, f: A→B is onto if for all y ∈B, there is
x ∈A such that f(x)=y.
• Example:
f: Z → Z
where for each x ∈ Z, f(x)=x^2.
• Clearly, f is NOT onto because 2 does not have any preimage.

Example onto function

• f: R → {0} ∪ R+
where for each x ∈ R, f(x)=x^2.
• f: N → Z+
where for each x ∈ N, f(x)=x+1.
• f:{0,1,2,3} →{0,1,2}
where f(0)=1, f(1)=1, f(2)=2, f(3)=0.

Sum of functions

• Suppose f1, f2, …, fn are functions from A to R. The sum of f1, f2, …, fn is also a
function from A to R defined as follows:
(f1+ f2+…+ fn )(x)= f1(x)+ f2(x) +…+ fn (x).
• Example:
f, g: R → R
where for each x ∈ R, f(x)=x+5; g(x)= x-3.
 • Then, f+g is defined as (f+g) (x)= 2x+2.

Product of functions

• Suppose f1, f2, …, fn are functions from A to R. The product of f1, f2, …, fn is also a
function from A to R defined as follows:
(f1 f2 … fn )(x)= f1(x) f2(x) … fn (x).
• Example:
f, g: R → R
where for each x ∈ R, f(x)=x+5; g(x)= x-3.
 • Then, f+g is defined as (f g) (x)= x^2+2x-15.

Bijection

• A function is a bijection if it is both one-to-one and onto.
• Example:
 – Consider f: Z+ → Z+ where for each x ∈ Z +, f(x)=x^2. This is NOT a bijection because it is not onto.
 – Consider f: R → {0} ∪ R+ where for each x ∈ R, f(x)=x^2. This is NOT a bijection either, because it is not one-to-one.
 – Consider f: R+ → R+ where for each x ∈ R+ , f(x)=x^2. This is a bijection.

Inverse function

• Suppose f is a bijection from A to B. The inverse function of f is the function from B to A that assigns element b of B to element a of A if and only if f(a)=b.
 – We use f-1 to represent the inverse of f.
 – Hence, f-1(b)=a if and only if f(a)=b.
• Example:
Consider f: R + → R+ where for each x ∈ R+ , f(x)=x^2. Its
inverse function is g: R + → R+ where for each x ∈ R+ ,
g(x)= sqrt (x).

Example inverse function

• Consider f: R + → R+ where for each x ∈ R+ , f(x)=4x+3. What is f-1 ?

• Consider f: {0,1,2} → {0,1,2} where f(0)=1, f(1)=2, f(2)=0. What is f-1 ?

Function composition

• Suppose g is a function from A to B, and f is a function from B to C. Then the composition of f and g is defined as:
(f o g) (x)= f(g(x)).
• Example: Consider f: R → R where for each x ∈ R , f(x)=2x+3, and g: R → R where for each x ∈ R , g(x)=3x-2.
Then, (f o g) (x)= f(3x-2) = 2(3x-2)+3=6x-1

Example composition

• Consider f: {0,1,2} →{0,1,2} where f(0)=1, f(1)=2, f(2)=0; also consider g: {0,1,2} →{1,2,3}
where f(0)=1, f(1)=2, f(2)=3.What is g o f ?

Incommutability of composition

• Consider f: R → R where f(x)=x+1, and g: R → R where g(x)=x^2.
 – It is not hard to get that f o g (x) = x^2 +1.
 – Similarly, we have that g o f (x) = (x +1)^2.
 – Clearly, f o g ≠ g o f

• In general, function composition is incommutable, which means the order of arguments in composition is important.

Graph of function

• We can often draw a graph for a function f: A → B : For each x ∈ A, We draw a point (x, f(x)).
 – Typically, we need A and B to be subsets of R.
• The graph of some important functions:
 – Linear function f(x)= kx+b : line
 – Constant function f(x) = c: line parallel to the X-axis
 – Quadratic function f(x) = ax^2+bx+c: parabola

Sequence

• A sequence is a function whose domain is a set
of integers.
– The domain is typically Z+ (or, sometimes, N).
– The image of n is an.
– Each image an is called a term.
– For convenience, we often write it as a1, a2, …, or {an}.
• Example:
1, 4, 9, 16, 25,… is a sequence, where the nth term is a n =n^2 .

Example sequence

• 2, 9, 28, 65, … where the nth term is an =n^3 +1.
• 0, -2, -6, -12, … where the nth term is an = -(n-1)(n-2).
• 0, 1/2, 2/3, 3/4, … where the nth term is an = 1-1/n.
• -1, 1, -1, 1, … where the nth term is an = (-1)n.

Arithmetic sequence

• An arithmetic sequence is a sequence of the form a, a+d, a+2d, …
– Formally, it is a sequence {an}, where an =a+(n-1)d .
– Here a is called the initial term.
– Here d is called the common difference .
• Example: 9, 4, -1, -6, … is an arithmetic sequence, because it is of the form an = 9 -5(n- 1). The initial term is 9, and the common difference is -5.

Example arithmetic sequence

• Let x and y be two real numbers . Consider a sequence {an}, where an = 5xn +3y . Is this an arithmetic sequence?
 – The answer is yes, because we can rewrite it as an =(5x+3y)+5x(n-1).
 – The initial term is 5x+3y.
 – The common difference is 5x.

Geometric sequence

• A geometric sequence is a sequence of the form a, ar, ar2, …
– Formally, it is a sequence {an}, where an =arn-1 .
– Here a is called the initial term.
– Here r is called the common ratio.
• Example: 9, 3, 1, 1/3, … is an geometric sequence, because it is of the form an= 9 (1/3)n-1.
The initial term is 9, and the common ratio is 1/3.

Example geometric sequence

• Let x≠1 be a real number. Consider a
sequence {an}, where an =x2n+5 . Is this a
geometric sequence?
– The answer is yes, because we can rewrite it
as an =x^7 x^2n-2 = x^7 (x^2)n-1 .
– The initial term is x^7.
– The common ratio is x^2.

Sum of terms

• Given a sequence {an}, we can sum up its mth through nth terms. We write this sum as

– Note it is just a simplified way to write am+am+1+…an . There is no difference in meaning.
– Here i is called the index of summation, m is called the lower limit of the index, and n is called the upper limit.

Sum of arithmetic sequence

• We are often interested in the sum of first n terms of a sequence. For example, consider an arithmetic sequence {an},

where an =a+(n-1)d . We have

– So the main problem is to find

• What is

 – Let us denote it by S.
 –Recall its definition: S=1+2+…+n .
 –We can change order of terms: S= n+…+2+1.
 – Adding up the above two equations , we get 2S=(n+1)+(n+1)+…+(n+1)=n(n+1)
 –We get S=n(n+1)/2

• Now we come back to the sum of first n terms of arithmetic sequence:

• The above is the important formula for the sum of arithmetic sequence. You should memorize it.

Example sum of arithmetic sequence

• Consider the arithmetic sequence 9, 4, -1, -6, … The sum of first n terms is

• Thus the sum of first 10 terms is

Sum of geometric sequence

• Now consider an geometric sequence {an}, where an =arn-1 . We have

– So the main problem is to find

• What is

– Let us denote it by S.
– Recall its definition: S=1+R+…+rn-1 .
– We multiply both sides by r: rS= R+…+rn-1+rn.
– Taking the difference of two equations, we get (r-1)S=rn-1
– Assuming r≠1, we get S=(rn-1)/(r-1) .

• Now we come back to the sum of first n terms of geometric sequence:

• The above is the important formula for the sum of arithmetic sequence. You should memorize it.

• Bear in mind that this formula assumes r≠1.

(Question: what is the formula for r=1 ?)

Example sum of geometric sequence

• Consider the geometric sequence 9, 3, 1, 1/3, ... The sum of first n terms is

• Thus the sum of the first 5 terms is

Homework 5

• Due in class Oct 21, Tuesday
• Rosen 2.3: Questions 10, 14, 18, 58, 60.
• Rosen 2.4: Questions 2, 4, 14, 16, 18.
• Optional Question (Extra Credit of 10 points):
Consider a sequence {an}. Denote by Sn the sum
of first n terms of this sequence. Suppose that,
for all integer n≥2, we have that an+2SnSn-1=0.
Also suppose that a1=1/2.
(1) Prove that {1/Sn} is an arithmetic sequence.
(2) What is an ? Justify your answer

Prev Next