Value of Constant when continuous?

Question

For what value of the constant c is the function f continuous on ( -inifity , infitiy ) where 

 

f(x) = {x^2-c     if x ξ (-infinity, 9)

          {cx+7      if x ξ [9, infinity)

 

c= ????

Answer 1

Both formulas for f(x) are continuous over their respective subsets of the domain of f(x).  In order for f(x) to be continuous at x = 9, the limits of f(x) as x --> 9 from above and below have to be equal to each other.

 

The limit from below is 9² - c = 81 - c.  The limit from above is (c)(9) + 7 = 9c + 7

 

So 81 - c = 9c + 7

74 = 10c

7.4 = c