solve the function

Question

Solve the function

f(x)=(6x-6)(x+3)/(-6x-2)(3x-6)

What are the vertical asymptotes? Give a list of the x-values of the asymptotes separated by commas.

x=

what are the horizontal asymptote?

y=

What are the x-intercepts? Give a list of the x-values of the x-intercepts separated by commas.

x=

what is the y-intercept

y=

 

 

 

Answer 1

Vertical asymptode and values of x when (fx) is undefined
 
If -6x -2 = 0 , f(x) is undefined
 
i.e. at x = -1/3, f9x) will be undefined - first asymptode
 
Similarly if 3x-6 =0 , f(x) will be undefined
 
i.e at x = 2, f(x) will be undefined - second asymptode
 
Two asymptodes = -1/3, 2
 
X intercept when y or f(x) is zero
 
f(x) = 0 when either (6x-6) =0 or (x+3)=0 i.e. x=1 and x= -3 are x intercepts
 
Y intercept when x = 0
 
f(x) = y = (0-6)(0+3)/(0-2)(0-6) = -18/12 = - 1.5
y= -1/5 is y intercept
:)

Answer 2

Assuming:

 

f(x) = ( (6x-6)(x+3) )/( (-6x-2)(3x-6) )

 

f(x) = ( 6(x-1)(x+3) )/( (-2)(3x+1)(3)(x-2) )

 

f(x) = ( 6(x-1)(x+3) )/( (-6)(3x+1)(x-2) )

 

f(x) = -( (x-1)(x+3) )/( (3x+1)(x-2) )

 

There are no common factors in numerator and denominator; so no holes.

 

There are vertical asymptotes when 3x+1=0 and x-2=0; i.e., at x = -1/3 and x = 2.

 

f(10^100) ≈ -(10^100)(10^100)/((3*10^100)(10^100)) ≈ -1/3

 

f(-10^100) ≈ -(-10^100)(-10^100)/((-3*10^100)(-10^100)) ≈ -1/3

 

So there's a horizontal asymptote at y = -1/3.

 

x-intercepts occur when y-coordinate is zero:

 

0 = -( (x-1)(x+3) )/( (3x+1)(x-2) )

 

=> 0 = (x-1)(x+3)

 

So x-intercepts are x = 1 and x = -3.

 

The y-intercepts occur when x = 0:

 

y = -( (0-1)(0+3) )/( (3*0+1)(0-2) )

 

y = 3/(-2) = - 3/2

 

Here's a GeoGebra sketch:

http://www.dropbox.com/s/p8h161cd4cmf75f/sks23cu14020701.png