Chapter 1FunctionsLimitsand1.2 Functions
Chapter 1 Functions and Limits 1.2 Functions
I.Functionsf1. DefinitionA functionfis a rule of correspondence that associateswith each object x in one set and a single value yfromasecond set.VxEDEyERy= f(x)DomainRangeindependentvariabledependent variableS1.2 Functions
§1.2 Functions I. Functions 1. Definition A function f is a rule of correspondence that associates with each object x in one set and a single value y from a second set. x D y R y f (x) f ⎯→ = Domain Range independent variable dependent variable f
I. Functions1. DefinitionQ: how to find the domain of a function?1Ji = 3-x + arctanJz = VsinxxFind the circumference of a polygon with n equal sidesthat is inscribed to a circle with radius r.Q:how to discriminatetwo functions arethe same?f(x) =lgx2 , g(x)= 2lgxu= f(t) , y= f(x)S1.2Functions
§1.2 Functions Q: how to find the domain of a function? x y x 1 1 = 3 − + arctan y sinx 2 = Q: how to discriminate two functions are the same? f (x) lg x , g(x) 2lg x 2 = = u = f (t) , y = f (x) Find the circumference of a polygon with n equal sides that is inscribed to a circle with radius r. I. Functions 1. Definition
I. Functions2.SpecialFunctionsVGreatestintegerfunction[x]= the greatest integer less than or equal to x.0x1, x>0y0,x = 0Signfunctiony=sgnx=-1,x<00xx = sgnx·x$ 1.2Functions
§1.2 Functions 2. Special Functions Greatest integer function [x] = the greatest integer less than or equal to x. o x y Sign function − = = = 1, 0 0, 0 1, 0 sgn x x x y x x = sgn x x o x y I. Functions
f(x)Il.PropertiesofFunctions→RD.1. BoundednessDef: We say f(x) is bounded above (below)onXifVx E X,3M > 0, suchthat f(x)≤ M(f(x)≥-M)yM0xNS1.2Functions
§1.2 Functions II. Properties of Functions 1. Boundedness D R ⎯f ⎯( x) → Def: x X,M 0, such that f (x) M( f (x) −M). We say f (x) is bounded above (below) on X if y x o M N