Functions

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Functions
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Definition and notation

• Definition A function f from a set X to a set Y
• is a relationship between elements of X and Y
• with the property that
• each element of X is related to a unique
  element of Y.
• Denoted fX?Y .
• X is called domain of f Y is called co-domain of
  f.
• Example XZ, YZ and f x ? 2?x / 2?
• ?x?X ?! y?Y such that f(x)y .
• f(x) is called f of x (or image of x under f).
• Range of f y?Y yf(x) for some x in X
• Inverse image of y x?X f(x)y
• Example(cont.) range of f all even integers
  
• inverse image of 4 3, 4 .

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Examples of Functions

• Squaring function f x ? x2 .
• Constant function f x ? 3 .
• Linear function f x ? 3x2 .
• Factorial function f n ? n! .
• Any sequence can be considered
• as a function defined on a set of integers.
• E.g., sequence 2,5,8,11,14,
• is a function from Z to Z
• defined as follows f n ? 3n-1

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Boolean Functions

• Recall the truth tables
• Can be considered as a function
• the domain is the set
• of all ordered couples of 0 and 1
• the co-domain is 0,1 .

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Boolean Functions

• Definition
• An (n-place) Boolean function is a function
• whose domain
• is the set of all ordered n-tuples of 0s and
  1s
• and whose co-domain
• is the set 0,1.
• Example f (x,y,z) ? (x ? y) ? z

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One-to-one Functions

• Definition
• Let F be a function from set X to set Y.
• F is one-to-one (or injective) iff
• for all elements x1, x2 ? X
• if F(x1)F(x2) then x1x2 .
• Examples Define f Z ? Z by f(n)2n3
• g R ? R by f(x)x2 .
• Then f is one-to-one, and g is not.

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Onto Functions

• Definition
• Let F be a function from set X to set Y.
• F is onto (or surjective) iff
• for any element y ? Y there is a x ?X
• such that F(x)y .
• Examples Define f Z ? Z by f(n)2n3
• g Z ? Z by f(n)n-2 .
• Then g is onto, and f is not.

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Exponential Functions

• The exponential function with base b
• is the following function from R to R
• expb(x) bx
• b01 b-x 1/bx
• bubv buv
• (bu)v buv
• (bc)u bucu

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Logarithmic Functions

• The logarithmic function with base b
• (bgt0, b?1)
• is the following function from R to R
• logb(x) the exponent to which b must
  raised to obtain x .
• Symbolically, logbx y ? by x .
• Properties

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One-to-one Correspondences

• Definition A one-to-one correspondence
• (or bijection) from a set X to a set Y
• is a function fX?Y
• that is both one-to-one and onto.
• Examples
• 1) Linear functions f(x)axb when a?0
• (with domain and co-domain R)
• 2) Exponential functions f(x)bx (bgt0, b?1)
• (with domain R and co-domain R)
• 3) Logarithmic functions f(x)logbx (bgt0, b?1)
• (with domain R and co-domain R)

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Inverse Functions

• Theorem
• Suppose F X?Y is a one-to-one correspondence.
• Then there is a function F-1 Y?X defined as
  follows
• Given any element in Y,
• F-1(y) the unique element x in X
• such that F(x)y .
• The function F-1 is called the inverse function
  for F.
• Example
• The logarithmic function with base b (bgt0, b ?1)
  
• is the inverse of the exponential function
  with base b.