Consider the function 𝑓 of 𝑥 equals two 𝑥 squared minus 𝑘𝑥 plus four for 𝑥 less than or equal to three and 𝑘 𝑥 squared minus 14 for 𝑥 greater than three. For what value of 𝑘 is 𝑓 continuous at 𝑥 equals three?
Looking at the function 𝑓 we can see that it is a piecewise function with 𝑥 equals three being the value of which its definition changes. The function 𝑓 will be continuous if the left-hand limit, that is, the value of the function as 𝑥 approaches three from below, is equal to the value of the function when 𝑥 equals three and is equal to the right-hand limit, the value of the function as 𝑥 approaches three from above. Now, the value 𝑥 equals three itself is included in the first part of the definition of 𝑓 of 𝑥. And as this part of the piecewise function is itself continuous, this tells us that the left-hand limit as 𝑥 approaches three will be equal to the value of the function itself when 𝑥 equals three.
We can find expressions for both the left-hand limit and the right-hand limit in terms of 𝑘. The left-hand limit as the function approaches three from below and the value of the function itself when 𝑥 equals three is two multiplied by three squared minus 𝑘 multiplied by three plus four. That’s 18 minus three 𝑘 plus four which simplifies to 22 minus three 𝑘. The right-hand limit of the function as 𝑥 approaches three from above is 𝑘 multiplied by three squared minus 14 which simplifies to nine 𝑘 minus 14.
For the function to be continuous, we need the two limits to be the same. So we can take our two expressions involving 𝑘 and set them equal to one another, giving us a linear equation to solve in order to determine the value of 𝑘. We first add three 𝑘 to each side of the equation, then add 14 to each side of the equation, and finally divide by 12. We’ve found then that the value of 𝑘, for which the given function 𝑓 of 𝑥 is continuous at 𝑥 equals three, is 𝑘 equals three.