unknown variable

Question

Let f(x) = { cx + 7 , if x <=5 and x^2 , if x>5. What is the value of c that will make f continuous at x = 5?

Answer 1

The function will be continuous when x = 5 if the two pieces of the graph fit together at x = 5.

 

This will happen if 5c+7 = 5²

 

                          5c+7 = 25

 

                              5c = 18

 

                                c = 3.6

 

 

Answer 2

In order for f(x) to be continuous at x=5, the two pieces of the function must be equal to each other when x=5.  This means the two pieces must intersect at x=5.  When they intersect, there are no gaps.   That is what it means to be continuous.

 

cx + 7 = x²

 

 

Plug in x=5 into the equation to solve for c.

 

5c + 7 = 5²

 

5c = 5² - 7

 

5c = 25 - 7

 

5c = 18

 

  c = 3.6