# Video: APCALC03AB-P1A-Q04-727163738018

Consider the function π(π₯) = 2π₯ Β² β ππ₯ + 4 for π₯ β€ 3 and ππ₯ Β² β 14 for π₯ > 3. For what value of π is π continuous at π₯ = 3?

02:12

### Video Transcript

Consider the function π 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 π continuous at π₯ equals three?

Looking at the function π we can see that it is a piecewise function with π₯ equals three being the value of which its definition changes. The function π 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 π 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. Thatβ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. Weβve found then that the value of π, for which the given function π of π₯ is continuous at π₯ equals three, is π equals three.