compare and contrast the functions

Question

graph the functions 

f(x)=(-x^3+5x^2-6x-12)/(x^2-2x) and

g(x)=-x+3

 

what are the similarities between f(x) and g(x)? please choose which ones apply 

1)they have the same end behavior

2)they have the same x-intercepts 

3)they have the same y-intercepts 

4) they have the same asymptotes 

Answer 1

graph the functions

 

f(x)=(-x^3+5x^2-6x-12)/(x^2-2x) and
g(x)=-x+3

 

See GeoGebra graph here:

http://www.wyzant.com/resources/files/261712/slant_asymptote

What are the similarities between f(x) and g(x)? please choose which ones apply
1)they have the same end behavior   YES
2)they have the same x-intercepts     NO
3)they have the same y-intercepts     NO
4) they have the same asymptotes     NO; well, g is the slant asymptote of f, but g doesn't have any asymptotes because it's a line.

 

Let's confirm the slant asymptote of f by long division:

 

              -x+3                   

x^2-2x | -x^3+5x^2-6x-12

              -x^3+2x^2

                       3x^2

                       3x^2-6x

                                    -12

 

f(x) = -x+3-12/(x(x-2))

 

x → ±∞, f → (-x+3)^-; i.e., f approaches g from below.

 

Vertical Asymptotes at x = 0 and x = 2 (b/c can't divide by zero)

Answer 2

Hi Dalia,

            The only commonality between f(x) and g(x) is their asymptotes. As x gets large f(x) is very close to x and likewise g(x) is approximately equal to x also.

             This behavior can most easily be seen by plotting a few points of each function starting at x= 20 or so.

 

Regards

Jim