Act 10 Algebra Trabajo Colaborativo

1. De la siguiente relación R = {(x, y) / 3y + 〖4x〗^2 – 4x + 3 = 0}. Determine:

a) Dominio = todos los reales desde – infinito

3y + 〖4x〗^2– 4x + 3 = 0

3Y +〖4X〗^2- 4X + 3 = 0

Y= - 4/3 ( x^2 -x+ 3/4 )

b) Rango.

A hora despejamos la x:

3y+〖4x〗^2-4x+3=0

〖ax〗^2+bx+c=0 a ≠0

〖4x〗^2-4x+3+3y=0

a=4

b=-4

c= 3+3y

x=- (b+ - √(b^2-4ac))/2a

x=(-(-4)±√((-4)^2-4(4)(3+3y)))/2(4)

x=(4±√(16-16(3+3y)))/8

16-16(3+3y)≥0

16-48-48y≥0

-32-48y≥0

-48y≥32

-y≥32/48

-y≥2/3

y≤-2/3

RANGO=(-∞,-2/3)

2. Dada las funciones f (x)= 3x - 2; g (x) = x3. Determine:

a) (f + g)(2) b) (f - g) (2) c) (f g) (2) d) (f / g) (2)

(〖F+G)〗_2= (3X - 2) + X^3

= -2 + 3X + X^3

=(2)^3+3(2)-2

= 8+ 6 – 2

=14-2

=12

(f-g)_2=(3x-2)-(x^3 )

=3x-2-x^3

=-x^3+3x-2

=-(2)^3+3(2)-2

=-8+6-2

= -4

(f*g)_2=(3x-2)(x^3 )

=3x^4-〖2x〗^3

=3(2)^4-〖2(2)〗^3

=3(16)-2(8)

=48-16

=32

(f/g)_2 =((3x-2))/((x^3 ) )

=((3(2)-2))/(((2)^3 ) )

=(6-2)/8

=4/8

= 2/4 = 1/2

3. Verifique las siguientes identidades:

a) (sec x + tan x) (1 – sen x) = cos x

b) tan⁡〖x+cos⁡x 〗/senx = sec x + cot x

(sec⁡x⁡〖+tan⁡x 〗 )(1-sin⁡x )=cos⁡x primero:

(sec⁡x⁡〖+tan⁡x 〗 )=cos⁡x/((1-sin⁡x ) )

(sec⁡x⁡〖+tan⁡x 〗 )=cos⁡x/((1-sin⁡x ) )×((1+sin⁡x ))/((1+sin⁡x ) )

(sec⁡x⁡〖+tan⁡x 〗 )=(cos⁡x (1+sin⁡x ))/(1-sin⁡x )(1+sin⁡x )

(sec⁡x⁡〖+tan⁡x 〗 )=(cos⁡x (1+sin⁡x ))/(1-sin⁡x )(1+sin⁡x )

(1-sin⁡x )(1+sin⁡x )=1+sin⁡x-sin⁡x-〖sin〗^2⁡〖x =1-〗 〖sin〗^2⁡x

(sec⁡x⁡〖+tan⁡x 〗 )=(cos⁡x (1+sin⁡x ))/((1-〖sin〗^2⁡x ) )

luego

(sec⁡x⁡〖+tan⁡x 〗 )=(cos⁡x (1+sin⁡x ))/cos^2⁡x =((1+sin⁡x ))/cos⁡x

(sec⁡x⁡〖+tan⁡x 〗 )=1/cos⁡x +sin⁡x/cos⁡x

sec⁡〖x=1/cos⁡x 〗 , tan⁡〖x=(sinx )/cos⁡x 〗

(sec⁡x⁡〖+tan⁡x 〗 )=(sec⁡x⁡〖+tan⁡x 〗 )

b)

(tanx+cosx)/senx=secx+cotx

Tan x = sen x/ cos x

((sinx/cosx)+cosx)/sinx=secx+cotx

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