eser-funzioni

8/12/2019 eser-funzioni

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2/16


3/16

|x 1|2

|2x + 1|


4/16

f(1) = arccos (log 1) = arccos (0) = 2

f(10) = arccos(log 10) = arccos (1) = 0

f(x)

f(1) = 2

f(2) =3 y= mx + q

m, q 2 =1m + q3 = 2m + q

2 =1m + q

5 =3m

m=53q= 2 53 = 13

y =53

x + 13

5x + 3y 1 = 0

f(x)

f(0) = 1, f(1) =0, f(3) = 5

y = ax2 +bx+c

a,b,c

f(0) = 1

f(1) = 0f(3) = 5

1 =c

0 =a + b + c5 = 9a + 3b + c

a=1 b

9 9b + 3b= 4c= 1

a= 76

b=136

c= 1

y= 76x2 13

6x + 1

f(4) =

2 f(5) = 6 f(4, 3)

f(x)

4x5

A(4; 2)

B(5; 6)

y y1y2 y1 =

x x1x2 x1

y+ 2

6 + 2 =

x 45 4

y+ 2

8 =x 4

y= 8x 34

f(4, 3)

x= 4, 3

y= 8 4, 3 34 = 0, 4

y= x24x3x x

3 x= 0 x= 3

R {} (; 3) (3;+)


5/16

y= x2xx24x+4

y = x2 x(x 2)2

x= 2

R {} oppure (; 2) (2;+)

y = x3+2xx23x+2 x = 1 x = 2

R {;} oppure (; 1) (1; 2) (2;+) y = x + 1

x + 10 x 1

[1; +)

y= 3x + 1

(; +) y= 1

4x2

x=2(; 2) (2; +2) (2;+)

y= x2 2 x2 20 x 2 x

2

;

2

2; +

y= 2 + x x2

x2 + x + 20x1,2=

11+82 = 2; 1

1x2

[1;2] y = x+2

xx+2

x x + 2 = 0

x + 2 0

x2 x 2 = 0x 2

x= 2 x=1

x 2

[2; 1) (1;2) (2;+) y= x + 1

2+x

x0 x02 + x >0 x >2

(2;0]


6/16

y =x+3x1

x + 30

x= 1

x 3x= 1

]3;1) (1;+)

y= x4 1 14

4

x4 10

(x

1) (x + 1) x

2 + 1 0

2

1 f attore x12 f attore x 1

x 1 x1(; 1[ ]1; +)

y=

1 x 1

1

x

1

0

x 10

x

1

1

x1 x2x1 CE: [1; 2 ]

y= lg

2+x2x

2 + x

2 x >0

N >0 x >2D >0 x 0

ln2 x 4= 0

x >0

ln2 x= 4

x >0ln x=2

x >0x=e2, 1

e2

0;

1

e2

1

e2; e2 e2; +


7/16

y= arccos 2x1+x

1 1

1 2x1+x

1x + 1= 0

2x1+x

12x1+x

1x=1

3x+11+x

0x11+x

0x=1

3x+11+x 0 N 0 x 1

3D >0 x >1 x 0 x >1 1< x1

13 1

[ 13 ; 1] y= sin2x

sin0 2k2x + 2k kx + k

f(x) = 2x4 3x3 5x2 + 6x 10

(x) = 12

[f(x) + f(x)] e (x) =12

[f(x) f(x)]

(x) = 1

2

2x4 3x3 5x2 + 6x 10 + 2x4 + 3x3 5x2 6x 10

= 1

2

4x4 10x2 20

= 2x4 5x2 10

(x) = 1

2

2x4 3x3 5x2 + 6x 10 2x4 3x3 + 5x2 + 6x + 10

= 1

26x3 + 12x

= 3x3 + 6x

f(x) = (x) + (x)

f(x)

l < x < l

f(x) = f(x) f(x) =f(x)

f(x) = 12

(ax + ax)

f(x) = 12

(ax + ax)

f(x) = f(x)

f(x) = 1 + x + x2 1 x + x2 f(x) = 1 x + x2 1 + x + x2 f(x) = f(x)

f(x) = lg 1+x1x f(x) = lg 1x1+x = lg

1+x1x

1= lg 1+x

1x =f(x)

f(x) l < x < l

f(x) = (x) + (x) = 1

2[f(x) + f(x)] +1

2[f(x) f(x)]

(x) = 12

[f(x) + f(x)]

(x) = (x) = 12

[f(x) + f(x)]

(x) = 12

[f(x) f(x)] (x) = 12[f(x) f(x)] = (x)

f(x) = (x) + (x)

pari dispari

y= M N

S

AM N

x= AM AB= a BC=b AH=c


8/16

AHD

AMN

AH : BC = AM :MN c: b = x : y

y= bc

x

b > c

S= x bc

x 12

= b

2cx2

0xc

y(x)

bc

S(x) V(0;0)

AB= l

q1, q2, q3

AC=l1

CD = l2

DB = l3

l1 + l2 + l3= l

m AM=x x

q1=m1

l1q2=

m2l2

q3= m3

l3

0xl1 q1= mx m= q1x x= 0 m = 0 x= l1m = m1

l1xl1+ l2 m= q1l1+ q2(x l1) =q1l1+ q2x q2l1= q2x + l1(q1 q2) x= l1

m = m1 x= l1+ l2

m = m2+ m1

l1+ l2xl1+ l2+ l3 m= q1l1+ q2l2+ q3(x l1 l2) =q3x + l1(q1 q3) + l2(q2 q3)

x= l1+ l2 m = m1+ m2 x= lm = q3l1+ q3l2+ q3l3+ q1l1 q3l1+ q2l2 q3l2=m1+ m2+ m3

x, m

[ (x)]

[ (x)]

(x) = x2

(x) = 2x

[ (x)] = (2x)2 = 22x [ (x)] = 2x2

f{f[f(x)]}

f(x) = 11x

f[f(x)] = 11 1

1x

= 11x11x

= 1 x f{f[f(x)]}= 1 1 + x= x

f(x + 1)

f(x 1) = x2

f(x) = (x + 1)2 f(x + 1) = (x + 2)2


9/16

y

y= x + 1 y >0 x + 1> 0 1< x 0 3< x 31

x 11 + x

>0

N >0 x >1D >0 x >1

y >0 x 1 y 0

y= (2x 5)10

u= 2x 5

y= u10

y= 2cosx u= cos x y= 2u

y= lg tan x2

u= x2

v= tan u

y = lg v

y= arcsin

3x2

u= x2 v= 3u y= arcsin v

y= u2

u= sin x

y = sin2 x

y= arctan u u=

v v= lg x y = arctan lg1

2 x

y=

2u se u00 se u >0

u= x2 1

y=

2

x2 1 se 1x10 se x 1

y


10/16

x2 arccos y = arccos y = x2 y = cos x2 = cos x2

y=cos x2

x+|y|= 2y

x + y= 2y se y0x y= 2y se y


11/16

x 1, 3 1, 4

xy = 6

x2 x + y = 4

y =x2 +x + 4

V( b2a

= 12

4a

= 174

x2


12/16

y = mx+ q

m

q

q

y

m

m= y2y1x2x1 x1, x2 y1, y2

q= 3 Q(0; 3)

m

m


13/16

m

y=3x + 3

y =kx k= 1, 2,

1,

2 q= 0

m < 0

y= x + b b= 1, 2, 1, 2 b

y


14/16

y = ax2 a = 1, 2, 1, 2 a

x, y

y = x2 +c c = 1, 2, 1

c


15/16

y= x3

y= 2 + (x 1)3

(1, 0)

(x 1)3

(0, 2)

y = x3


16/16

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