SOLUTION: Find the values of a and b that will make the function continuous every where. Justify. f(x)= x^2+11x+10/x+1, x<-1 e^2ax , -1<= x <= 1 bx^2-ax, x>1 thank you in advance!

Algebra ->  College  -> Linear Algebra -> SOLUTION: Find the values of a and b that will make the function continuous every where. Justify. f(x)= x^2+11x+10/x+1, x<-1 e^2ax , -1<= x <= 1 bx^2-ax, x>1 thank you in advance!       Log On


   

Question 1017899: Find the values of a and b that will make the function continuous every where. Justify.
f(x)=
x^2+11x+10/x+1, x<-1
e^2ax , -1<= x <= 1
bx^2-ax, x>1
thank you in advance!
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I believe what you meant is
f%28x%29%22=%22

The pieces of that function are continuous in
%22%28%22-infinity%22%2C%22%22-+1+%29%22 , %22%5B%22-1%22%2C%22%221+%29%22 , and %22%28+1%22%22%2C%22infinity%22%29%22 .
We just have to make it continuous at x=-1 and x=1 .

For x=-1 , we have f%28-1%29=e%5E%28-2a%29 .
For x%3C-1 , f%28x%29=%28x%5E2%2B11x%2B10%29%2F%28x%2B1%29=%28x%2B1%29%28x%2B10%29%2F%28x%2B1%29=x%2B10 , and
lim%28x-%3E-1%2Cf%28x%29%29=lim%28x-%3E-1%2Cx%2B10%29=-1%2B10=9 ,
so for f%28x%29 to be continuous at x=-1 ,
it must be e%5E%28-2a%29=9-->e%5E%28-2a%29=3%5E2-->-2a=2ln%283%29-->highlight%28a=-ln%283%29%29

For x=1 , we have f%281%29=e%5E%282a%29 .
For x%3E1 , f%28x%29=bx%5E2-ax , and
lim%28x-%3E1%2Cf%28x%29%29=lim%28x-%3E1%2Cbx%5E2-ax%29=b-a ,
so for f%28x%29 to be continuous at x=-1 ,
it must be b-a=e%5E%282a%29 .
Since we knew that
system%28e%5E%28-2a%29=9%2C%22and%22%2Ca=-ln%283%29%29-->system%28e%5E%282a%29=1%2F9%2C%22and%22%2Ca=-ln%283%29%29 ,
substituting into b-a=e%5E%282a%29 , we get b%2Bln%283%29=1%2F9-->highlight%28b=ln%283%29%2B1%2F9%29