calculus- continuous limits

Question

f(x)=9x^-1 for x<-1

     =ax+b for -1≤x≤0.5

     =5x^-1 for x>0.5

what are the values of a and b to make the function continuous

Answer 1

For this piecewise function f(x) to be continuous at x = -1:

1. f(-1) must exist
2. lim_x->-1f(x) must exist (which implies limit from the left equals the limit from the right)
3. f(-1) = limit as x approaches -1

Basically that means that when x = -1, 9x^-1 must have the same value as ax + b.

9(-1)^-1 = a(-1) + b

a - b = 9

Applying similar logic as x -> 1/2:

5(1/2)^-1 = (1/2)a + b

10 = (1/2)a + b

20 = a + 2b

a + 2b = 20

a - b = 9

----------------------

3b = 11

b = 11/3

a = 9 + 11/3 = 38/3

Visit this Desmos graph and click on the "Make f continuous" link to see what the graph looks like when a and b take on the above values:

desmos.com/calculator/2r1yzggwtk