Video: Redefining a Piecewise Defined Function to Make It Continuous at a Point

Video Transcript

Given that the function 𝑓 of 𝑥 is equal to negative seven 𝑥 plus eight if 𝑥 is less than negative eight, and 𝑓 of 𝑥 is equal to 𝑥 cubed plus two 𝑥 plus four if 𝑥 is greater than negative eight. If possible or necessary, define the function 𝑓 evaluated at negative eight so that 𝑓 is continuous at 𝑥 is equal to negative eight.

We can see that the question is asking us to try to make the function 𝑓 continuous when 𝑥 is equal to negative eight. And it asks us to do this by defining the function 𝑓 evaluated at negative eight, but only if this is possible or necessary. The first thing we should do is recall what it means for a function to be continuous at a point 𝑥 is equal to 𝑎. We say that the function 𝑓 is continuous when 𝑥 is equal to 𝑎 if the following criteria hold. First, we must have that 𝑓 is defined at 𝑎 which, it’s worth noting, is equivalent to saying that 𝑎 is in the domain of our function 𝑓.

The second requirement is that the limit as 𝑥 approaches 𝑎 of our function 𝑓 of 𝑥 must exist. And then, once we’ve checked our first two requirements, our final requirement is that the limit as 𝑥 approaches 𝑎 of our function 𝑓 of 𝑥 must be equal to 𝑓 evaluated at the point 𝑎. Since the question is asking us about the continuity of our function 𝑓 of 𝑥 at the point 𝑥 is equal to negative eight, we can change all instances of 𝑎 in our definition of continuity with negative eight. We’re now ready to start checking each part of our definition of continuity. Let’s start with checking if 𝑓 is defined at 𝑥 is equal to negative eight.

To evaluate 𝑓 of negative eight, we would need to use the definition of our function 𝑓 of 𝑥 given to us in the question. However, we see the first segment in our piecewise definition has 𝑥 is strictly less than negative eight. And the second segment in our piecewise definition has 𝑥 is strictly greater than negative eight. Therefore, we can conclude that our function 𝑓 evaluated at negative eight is not defined. However, this is not the end of the question. Since we’re allowed to define the function 𝑓 evaluated at negative eight ourselves to try and make our function 𝑓 of 𝑥 continuous at the point 𝑥 is equal to negative eight.

So let’s move on to our second criteria for continuity. The limit of 𝑓 of 𝑥 as 𝑥 approaches negative eight must exist. We recall that the limit of 𝑓 of 𝑥 as 𝑥 approaches negative eight existing is the same as saying that the limit of 𝑓 of 𝑥 as 𝑥 approaches negative eight from the left and that the limit of 𝑓 of 𝑥 as 𝑥 approaches negative eight from the right exist. And that both of these limits are equal. So to check our second criteria, we must check the left- and right-hand limit of our function 𝑓 of 𝑥 as 𝑥 approaches negative eight. For our left-hand limit, we can see when 𝑥 approaches negative eight from the left, we must have that 𝑥 is less than negative eight.

From the question, we can see that when 𝑥 is less than negative eight, 𝑓 of 𝑥 is equal to the function negative seven 𝑥 plus eight. So we can replace the 𝑓 of 𝑥 in our left-hand limit with negative seven 𝑥 plus eight. Since we’re now trying to evaluate the limit of a linear function, we can use direct substitution. This gives us that our left-hand limit is equal to negative seven multiplied by negative eight plus eight. Which we can evaluate to give us an answer of 64.

We can now do the same thing for our right-hand limit. We see that when 𝑥 approaches negative eight from the right, we must have that 𝑥 is greater than negative eight. We can see from the definition given in our question that when 𝑥 is greater than negative eight, our function 𝑓 of 𝑥 is equal to the function 𝑥 cubed plus two 𝑥 plus four. Therefore, we can replace the 𝑓 of 𝑥 in our right-hand limit with 𝑥 cubed plus two 𝑥 plus four. Since we are now trying to evaluate the limit of a polynomial, we can do this by using direct substitution.

Substituting in 𝑥 is equal to negative eight gives us negative eight cubed plus two multiplied by negative eight plus four, which, if we calculate, gives us an answer of negative 524. If we were to sketch a graph of our function 𝑓 of 𝑥, as defined in the question, it would look like this. We can see from our sketch that when 𝑥 approaches negative eight from the left, we get closer and closer to 64. And we can also see that as 𝑥 approaches negative eight from the right, 𝑓 of 𝑥 is getting closer and closer to negative 524.

We can see from our sketch or from the second criteria in our definition of 𝑓 being continuous that us defining 𝑓 of negative eight will not change whether the left- and right-hand limit are equal. Therefore, we can conclude that the function 𝑓 of 𝑥 cannot be made continuous at the point 𝑥 is equal to negative eight because the left-hand limit and the right-hand limit of 𝑓 of 𝑥 as 𝑥 approaches negative eight are not equal.