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
f(x)Il.PropertiesofFunctionsRD.1. BoundednessDef: We say f(x) is bounded above (below)onXifVx E X,3M > 0, suchthat f(x)≤ M(f(x)≥-M)We say f(x) is bounded on XifIf(x)/≤ M.Vx E X,3M > 0, such thatMxMS1.2 Functions
§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 x X,M 0, such that f (x) M. We say f (x) is bounded on X if -M y x o M N
f(x)II. Properties of FunctionsRD2. MonotonicityDef: X c D, Vxi,x, E X, Xi f(x,)10n (0,+)Forexample,y=x0n (-00,0) U (0,+0)S1.2 Functions
§1.2 Functions 2. Monotonicity then f is increasing on X if ( ) ( ) 1 2 1 2 x x f x f x f is decreasing on X if ( ) ( ) 1 2 1 2 x x f x f x Def: , , , , X D x1 x2 X x1 x2 For example, on (−,0)(0,+) on (0, ) 1 = + x y II. Properties of Functions D R ⎯f ⎯( x) →
D(x)→RII.Properties of Functions3. Odd and even functionsDef: Let D be symmetric to the origin.We say f(x) is an odd function if Vx D, f(-x) =-f(x)We say f(x) is an even function if Vx E D, f(-x) = f(x)4. Periodic functionsDef: we say f(x) is a periodic function ifl,Vx E D(x±l e D) such that f(x+l)= f(x)Forexample, y = sinx,y = cosxS1.2Functions
§1.2 Functions II. Properties of Functions D R ⎯f ⎯( x) → 3. Odd and even functions For example, y = sinx, y = cos x 4. Periodic functions Def: l,x D ( x l D) such that f (x + l) = f (x). we say f (x) is a periodic function if Def: Let D be symmetric to the origin. We say f (x) is an even function if x D, f (−x) = f (x). We say f (x) is an odd function if x D, f (−x) = − f (x)
Ill.InverseFunctionsVxED-/3yERy= f(x)x= f-l(y) y=f-l(x)VyER3xED<xy= f(x)y=xy=f-l(x)EO(a,b)x=f-l(y)(b,a)0yx0DOMAIN OFf-1DOMAINOFfIThegraphs of f andf-'(x)aresymmetricwithliney=x. If f is a monotonic function, then so is f-l(x).S1.2Functions
§1.2 Functions y = f (x) III. Inverse Functions x D y R y f (x) f ⎯→ = x D ⎯⎯ y R ( ) 1 x f y − = ( ) 1 y f x − = ◼The graphs of f and are symmetric with line ( ) y =x. 1 f x − ◼ If f is a monotonic function, then so is ( ). 1 f x −
IV. CompositeFunctionsy= f(u)u= g(x)y = f[g(x)]=(f og)(x)D, RD, R(f o g)(x)is called the composition of fwith g.Restriction:u = g(x)is in the domain of f,that is R cDFor example: y = V4-x?.2u=2+xy = arcsinu,S1.2 Functions
§1.2 Functions IV. Composite Functions y = f (u) u = g(x) 1 1 D, R D , R y = f[g(x)] = ( f g)(x) ( f g)(x) is called the composition of f with g. Restriction: u = g(x) is in the domain of f ,that is . R1 D For example: 2 y = 4 − x 2 y = arcsinu, u = 2+ x