If f ( x ) is a function which is continuous everywhere, then we must have

Question

Let f(x)={8x−2 ifx ≤ 10

{−8x+b if x>10

If f ( x ) is a function which is continuous everywhere, then we must have

b = ?

Answer 1

Hi Abraham,

The function:

f(x)= {8x − 2 if x ≤ 10

{−8x + b if x > 10

is not continuous. We can prove this by finding each of the one-sided limits as x→10

lim_x→10- = 8x - 2

= 8(10) - 2

= 78

lim_x→10+ = -8x +b

= -8(10) + b

= b - 80

For the function to be continuous,

lim_x→10- = lim_x→10+

As x approaches 10 from the left, the function approaches 78, in order for the function to be continuous, the function must also approach 78 from the right.

The problem is asking us, "For what value of b, will the function be continuous?"

To solve for b, we can set f(x ≤ 10) = f(10 < x) to get,

8x − 2 = −8x + b

16x - 2 = b

substitute 10 for x

16(10) - 2 = b

158 = b

When b = 158, as the function approaches ten from both the left and the right sides, they approach the same value.

8x − 2 = −8x + b

8(10) - 2 = -8(10) + 158

78 = 78

Hope this helps!