Functions

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} .

Examples of Functions Squaring function: f : x → x2 . 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

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} . Input Output p q p  q 1

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

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 x1=x2 . Examples: Define f : Z → Z by f(n)=2n+3 ; g : R → R by f(x)=x2 . Then f is one-to-one, and g is not.

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.

Exponential Functions The exponential function with base b is the following function from R to R+ : expb(x) = bx b0=1 b-x = 1/bx bubv = bu+v (bu)v = buv (bc)u = bucu

Logarithmic Functions The logarithmic function with base b (b>0, 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:

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)=bx (b>0, b1) (with domain R and co-domain R+) 3) Logarithmic functions: f(x)=logbx (b>0, b1) (with domain R+ and co-domain R)

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.