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1 Functions

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2 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 f:X→Y. X is called domain of f; Y is called co-domain of f. Example: X=Z, Y=Z 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 | y=f(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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3 Examples of Functions Squaring function: f : x → x 2. Constant function: f : x → 3. Linear function: f : x → 3x+2. 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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4 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}. InputOutput pq p q

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5 Boolean Functions Definition: An (n-place) Boolean function is a function whose domain is the set of all ordered n-tuples of 0’s and 1’s and whose co-domain is the set {0,1}. Example: f : (x,y,z) → (~x y) z

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6 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 x 1, x 2 X if F(x 1 )=F(x 2 ) then x 1 =x 2. Examples: Define f : Z → Z by f(n)=2n+3 ; g : R → R by f(x)=x 2. Then f is one-to-one, and g is not.

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7 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)=2n+3 ; g : Z → Z by f(n)=n-2. Then g is onto, and f is not.

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8 Exponential Functions The exponential function with base b is the following function from R to R + : exp b (x) = b x b 0 =1 b -x = 1/b x b u b v = b u+v (b u ) v = b uv (bc) u = b u c u

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9 Logarithmic Functions The logarithmic function with base b (b>0, b 1) is the following function from R + to R: log b (x) = the exponent to which b must raised to obtain x. Symbolically, log b x = y b y = x. Properties:

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10 One-to-one Correspondences Definition: A one-to-one correspondence (or bijection) from a set X to a set Y is a function f:X→Y that is both one-to-one and onto. Examples: 1) Linear functions: f(x)=ax+b when a 0 (with domain and co-domain R) 2) Exponential functions: f(x)=b x (b>0, b 1) (with domain R and co-domain R + ) 3) Logarithmic functions: f(x)=log b x (b>0, 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 (b>0, b 1) is the inverse of the exponential function with base b.

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