Video: APCALC03AB-P1A-Q04-727163738018

Consider the function 𝑓(π‘₯) = 2π‘₯ Β² βˆ’ π‘˜π‘₯ + 4 for π‘₯ ≀ 3 and π‘˜π‘₯ Β² βˆ’ 14 for π‘₯ > 3. For what value of π‘˜ is 𝑓 continuous at π‘₯ = 3?

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Video Transcript

Consider the function 𝑓 of π‘₯ equals two π‘₯ squared minus π‘˜π‘₯ plus four for π‘₯ less than or equal to three and π‘˜ π‘₯ squared minus 14 for π‘₯ greater than three. For what value of π‘˜ is 𝑓 continuous at π‘₯ equals three?

Looking at the function 𝑓 we can see that it is a piecewise function with π‘₯ 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 π‘₯ approaches three from below, is equal to the value of the function when π‘₯ equals three and is equal to the right-hand limit, the value of the function as π‘₯ approaches three from above. Now, the value π‘₯ equals three itself is included in the first part of the definition of 𝑓 of π‘₯. And as this part of the piecewise function is itself continuous, this tells us that the left-hand limit as π‘₯ approaches three will be equal to the value of the function itself when π‘₯ equals three.

We can find expressions for both the left-hand limit and the right-hand limit in terms of π‘˜. The left-hand limit as the function approaches three from below and the value of the function itself when π‘₯ equals three is two multiplied by three squared minus π‘˜ multiplied by three plus four. That’s 18 minus three π‘˜ plus four which simplifies to 22 minus three π‘˜. The right-hand limit of the function as π‘₯ approaches three from above is π‘˜ multiplied by three squared minus 14 which simplifies to nine π‘˜ 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 π‘˜ and set them equal to one another, giving us a linear equation to solve in order to determine the value of π‘˜. We first add three π‘˜ 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 π‘˜, for which the given function 𝑓 of π‘₯ is continuous at π‘₯ equals three, is π‘˜ equals three.

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