Calculus Exercise Chapter 1 Function 1.Are they functions in the following graphs? rz☑ 4p→. 2.Are they functions in the following tables? d x111 x123 x111 )i23 111 x)11I e x123 x223 x123 x)456 x)111 )121o2
Calculus Exercise Chapter 1 Function 1. Are they functions in the following graphs? a) b) c) d) e) f) g) h) i) j) k) 2. Are they functions in the following tables? a) b) c) d) e) f) 0 20 40 60 80 100 1st season 2nd season 3rd season 4th season x 1 1 1 f(x) 1 2 3 x 1 2 3 f(x) 1 1 1 x 1 1 1 f(x) 1 1 1 x 1 2 3 f(x) 4 5 6 x 2 2 3 f(x) 1 1 1 x 1 2 3 f(x) 1 2 1or2
3.Are they functions in the following formulas? a)f(x)=sinx;b)f(x)=0. of()=f xo -x,otherwise; d f()=2 Ux=1 -2,fx=-l e)f(x)=y,where y satisfiesx2+y2=1; 「sinx,ifx<0, ffx)=e",x∈(0，+o),g)f(x)=L,fx=0,h)fx)= 2，fx≤0 0,fx20 x,if x>0. h)f(x)=cosl; i)fx)=g2(x),where g(x)=tanx,in(-π/2，π/2： Dr++4-2in0+啡为=sire)in(m以 sin(e*) 3.ifx<0, )f=snM儿haeM)-fx之0mf=； wo-hrm--2 2,ifx2>1. sr=me sine*-vx2+3x+I,ifx2+3x+1z0 =F不，刂-高-@gw wf)=log(sinx-ly为fx)=--可+1 4 y)f(x)=1+x+++x,where N is a positive integer; z)f(x)=1+sinx+sin 2x+..+sin Nx,where N is a positive integer; 4.Are they functions in the following sentences? a)f(n)denotes the average temperature in degree of the days since 01/12/2013; b)f(n)is denoted as the number of the students of the class n;
3. Are they functions in the following formulas? a) f x x ( ) sin ; b) f x( ) 0; c) , 0, ( ) , ; x if x f x x otherwise d) 2, 1, ( ) 2, 1; if x f x if x e) 2 2 f x y y x y ( ) , where satisfies 1; f) 1/ ( ) , (0, ); x f x e x g) sin , 0, ( ) 1, 0, , 0; x if x f x if x x if x h) 2, 0, ( ) 0, 0; if x f x if x h) f x( ) cos1; i) 2 f x g x g x x ( ) ( ), where ( ) tan , in ( / 2, / 2); j) 2 sin( ) ( ) , in (0, ); 4 2 x e f x x x k) 2 cos ( ) sin( ), in ( , ); x x e f x e l) 3, if 0, ( ) sin( ( )), where ( ) ; 3, if 0 x f x h x h x x m) f x e ( ) ; n) 1, if 1, ( ) 2, if 2; x f x x o) 2 2 1, if 1, ( ) 2, if 1, x f x x p) 1, if 1, ( ) 2, if 2; x f x x q) log cos , if cos 0, ( ) 1 cos , if cos 0; x x f x x x r) 2 2 2 2 sin 3 1, if 3 1 0, ( ) 1 log | | , if 3 1 0; 1 x e x x x x f x x x x x s) 2 f x x ( ) 1; t) 1 ( ) cos f x x u) f x x ( ) loglog ; v) f x x x ( ) sin w) f x x ( ) log(sin 1); x) 4 3 2 1 4 ( ) 1 ; 1 3 2 1 f x x x x x y) 2 ( ) 1 ... ,where is a positive integer; N f x x x x N z) f x x x Nx N ( ) 1 sin sin 2 ... sin ,where is a positive in teger; 4. Are they functions in the following sentences? a) f(n) denotes the average temperature in degree of the days since 01/12/2013; b) f(n) is denoted as the number of the students of the class n;
c)f(n)is denoted as the number of the students under 17 ages of the class n. d)f(n)is denoted as the number of positive results in n tests; gaBaaieashgpoIyPceoheoCkaaheomoommaTteB g)f(n)is denoted as the expected number of the good medals of the country n in the next Olympic Games. 5.Find the domains of the following functions: a)f(x)=0.b)f(x)=sinx;c)f(x)=2x2-x+3 d)f(x)=e+c0. )(log(1): 1 0-2-+2nf国=（可w2-- 1 f(x)=sin(log(x));m)f(x)=log(cosx)n)f(x)=-e' o=-cs/-画，g倒=e阿 r)f(x)=Vx2-3x+1:s)f(x)= =, V Ixl -1,ifx<I 0=x2M= x+l,ifx>-l, cosx,if-3<x<0; x,otherwise; [e,if2x-1≥0， w)fx)=e,if sinx2x刈f(x)= [logx,otherwise; 1,fx>1 v)f(x)=log(g(x)),where g(x)=0,if x=1. -1,ifx<l x123 x)321 6.Find the ranges of the following functions: a)fx)=上b)fx)=sinx3;dfx)=xe2；d)fx)=em
c) f(n) is denoted as the number of the students under 17 ages of the class n. d) f(n) is denoted as the number of positive results in n tests ; e) f(n) is denoted as the highest velocity of the car in type n; f) f(n) is denoted as the possibility price of the stock n in the tomorrow market; g) f(n) is denoted as the expected number of the good medals of the country n in the next Olympic Games. 5. Find the domains of the following functions: a) f x( ) 0; b) f x x ( ) sin ; c) 2 f x x x ( ) 2 3; d) 2 cos ( ) ; x x f x e e) 2 f x x ( ) log 1 ; f) 2 1 ( ) ; 3 1 f x x x g) 1 ( ) ; 1 x f x x h) 1 f x( ) ; x i) 2 1 ( ) ; 3 2 f x x x j) f x x x ( ) sin 1 ; k) 3 1 ( ) 3 ; 2 1 f x x x l) f x x ( ) sin log( ) ; m) f x x ( ) log(cos ) n) ( ) 1 ; x f x e o) 1 ( ) ; 1 cos f x x p) f x x ( ) tan ; q) 2 1 ( ) ; x f x e r) 2 f x x x ( ) 3 1; s) | 1| ( ) ; | | x f x x t) 2 2 1, if 2, ( ) 1, if 1; x f x x u) sin , if 1 2, ( ) cos , if -3< 0; x x f x x x v) 1, if 1, ( ) , otherwise; x x f x x w) ( ) , if sin 1; x f x e x x) , if 2 1 0, ( ) log , otherwise; x e x f x x y) 1, 1, ( ) log , where ( ) 0, 1, 1, 1. if x f x g x g x if x if x z) 6. Find the ranges of the following functions: a) f x( ) 1; b) 3 f x x ( ) sin ; c) 2 ( ) ; x f x xe d) 2 1 ( ) ; x f x e x 1 2 3 f(x) 3 2 1
=骨点N=F i)f(x)=xx2-1;i)f(x)=/cosx;k)f(x)=log(x+1);1)f(x)=tan 1f=闭=7o))=m2x+ s)f(x)=sece";t)f(x)=e;u)f(x)=logsinx+v)f(x)=e 1,ifx>0, w)f(x)=0,ifx=0,x)f(x)= +2x-1,ix2l0f)= x2+1,ifx=1, -1,ifx<0; e",otherwise; cosx,otherwise. x123 )321 7.Find the domains and ranges of the following functions by graph: b) 8.Indicate if the following functions are invertible or invertible after restrict the domains,if so find their inverse functions and their restricted domains if any: a)f(x)=0,b)f(x)=x,c)f(x)=x;d)f(x)=x+1;e)f(x)=2x-3, f)f(x)=/x+I;g)f(x)=x2-1;h)f(x)=x+x+1;i)f(x)=x;
e) sin ( ) ;x f x e f) 1 ( ) ; 1 x f x x g) 1 1 ( ) ; x x f x e h) 2 f x x ( ) 2; i) 2 f x x x ( ) 1; j) f x x ( ) cos ; k) f x x ( ) log( 1); l) f x x ( ) tan ; m) sin ( ) ; 1 x x f x e n) 1 ( ) ; 1 x x e f x e o) 2 f x x ( ) sin 1; p) 2 1 ( ) ; 1 f x x q) 1 ( ) ; cos 1 f x x r) 2 3 f x x x ( ) sin sin ; s) ( ) sec ; x f x e t) 2 cos ( ) ; x f x e u) f x x ( ) logsin 1; v) log ( ) ; x f x e w) 1, if 0, ( ) 0, if 0, 1, if 0; x f x x x x) 3 2 1, if 1, ( ) , otherwise; x x x x f x e y) 2 1, if 1, ( ) cos , otherwise. x x f x x z) 7. Find the domains and ranges of the following functions by graph: a) b) 1 0 1 8. Indicate if the following functions are invertible or invertible after restrict the domains, if so find their inverse functions and their restricted domains if any: a) f x( ) 0; b) f x x ( ) ; c) 3 f x x ( ) ; d) f x x ( ) 1; e) f x x ( ) 2 3; f) f x x ( ) 1; g) 2 f x x ( ) 1; h) 2 f x x x ( ) 1; i) 2 f x x ( ) 1 ; x 1 2 3 f(x) 3 2 1 1 0 1
)fx)=e+上kf(x)= F-3x+20f)=e:mf)=e2-+2 1 n)f(x)=e+2e*+l;o)f(x)=sine'-1;q)f(x)=esinx-; x,ifx≥0， f(x)x-1川+l)f(x)= 2,ix<0wf)=m≥0 -1,if x<0. sinx,ifx≥0， 1,if x21, x刘f(x)= tans.ifx<()if -l,ifx≤-l. x123 )456 9.Draw the inverse functions of the following functions from their graphs: 10.Value of the following functions: a)f(x)=sinx,thenf(0)= b)f(x)=2x2-1;thenf①= c)f(x)=cosx+e;thenf(z/2)=d)f(x)=x2-logx,thenf(1)= 。f)- ；thenf(π)= f)f(x)=xx+2;thenf(2)= g)f(x)=ve*+1;thenf(0)= h)f(x)=log(x2)thenf(ve)= i)f(1)=2,g(2)=3,theng(f(1))= k)f⑩=2，g(2)=2,theng(g(2)=
j) ( ) 1; x f x e k) 2 1 ( ) ; 3 2 f x x x l) 1 ( ) ; x f x e m) 2 1 ( ) 2; x f x e n) 2 ( ) 2 1; x x f x e e o) 3 ( ) sin 1; x f x e q) sin 1 ( ) ; x f x e r) ( ) log(1 cos ); x f x e s) 1 ( ) ; 1 x x e f x e t) f x x ( ) loglog( 1); u) f x x ( ) | 1| 1; v) 3 , if 0, ( ) , if 0. x x f x x x w) 1, if 0, ( ) 1, if 0. x f x x x) sin , if 0, ( ) tan , if 0. x x f x x x y) 1, if 1, ( ) , if -1< 1, 1, if 1. x f x x x x z) 9. Draw the inverse functions of the following functions from their graphs: a) b) 1 0 1 10. Value of the following functions: a) f x x f ( ) sin ; then (0) b) 2 f x x f ( ) 2 1; then (1) c) ( ) cos ; then ( / 2) x f x x e f d) 2 f x x x f ( ) log ; then (1) e) sin ( ) ; then ( ) 1 x f x f x f) f x x x f ( ) 2; then (2) g) ( ) 1; then (0) x f x e f h) 2 f x x f e ( ) log( ); then ( ) i) f g f (1) 2, g(2) 3, then ( (1)) k) f g g (1) 2, g(2) 2, then ( (2)) x 1 2 3 f(x) 4 5 6 1 0 1