Without using a calculator, find the correct value of k that makes the function f(x) continuous on [0, 11], if f is defined as follows:
Difficulty:
In order for f to be continuous, it can't have a break in the graph when x = 2. Therefore, you have to get the same output from both pieces of the function if you plug in 2 for x--matching heights ensures a connected graph. Let's start with the x > 2 rule; you can't just plug in 2 for x, because you'll get an indeterminate answer (0 divided by 0), so instead, calculate the limit of that function as x approaches 2 via the conjugate method of finding limits:
This tells us that the the other function, with 2 plugged in for x must also equal 1/6, so write that as an equation and solve for k:
With k equal to 1/3, both functions give you a value of 1/6 when you plug in 2 for x, so the function f is continuous. Notice that f is no longer continuous once x gets bigger than 11, because it results in a negative inside the radical.
