Question Video: Determining the Domain of a Piecewise-Defined Function Mathematics

Determine the domain of the function 𝑓(π‘₯) = π‘₯ + 4 when π‘₯ ∈ [βˆ’4, 8] and 𝑓(π‘₯) = 7π‘₯ βˆ’ 63 when π‘₯ ∈ (8 to 9].

02:02

Video Transcript

Determine the domain of the function 𝑓 of π‘₯ is equal to π‘₯ plus four when π‘₯ is in the closed interval from negative four to eight and 𝑓 of π‘₯ is equal to seven π‘₯ minus 63 when π‘₯ is in the left-open, right-closed interval from eight to nine.

In this question, we’re asked to find the domain of a given piecewise function 𝑓 of π‘₯. And we can start by recalling the domain of any function is the set of input values for that function. And for a piecewise-defined function 𝑓 of π‘₯, each sub function has its own subdomain. And in this case, 𝑓 of π‘₯ has two subfunctions: 𝑓 of π‘₯ is equal to π‘₯ plus four whenever π‘₯ is in the closed interval from negative four to eight and 𝑓 of π‘₯ is equal to seven π‘₯ minus 63 whenever π‘₯ is in the left-open, right-closed interval from eight to nine. Therefore, all of the input values of π‘₯ are defined by the subdomains. The union of the subdomains of a piecewise-defined function is the domain of our function. So, the domain of 𝑓 of π‘₯ is the union of its two subdomains.

And there’s a few different ways of finding the union of these two sets. First, to say that π‘₯ is in the closed interval from negative four to eight is the same as saying π‘₯ is greater than or equal to negative four and π‘₯ is less than or equal to eight. We can say the same for our other subdomain. To say that π‘₯ is in the left-open, right-closed interval from eight to nine is the same as saying that π‘₯ is greater than eight or less than or equal to nine. To say that π‘₯ is in the union of these two sets is the same as saying either of these two conditions are true. And saying that π‘₯ is greater than or equal to negative four and less than or equal to eight or π‘₯ is greater than eight and less than or equal to nine is just the same as saying that π‘₯ is between negative four and nine.

And since our inequalities at the end of our expression are not strict, we need to use a closed interval. The domain of the function 𝑓 of π‘₯ is equal to π‘₯ plus four when π‘₯ is in the closed interval from negative four to eight and 𝑓 of π‘₯ is equal to seven π‘₯ minus 63 when π‘₯ is in the left-open, right-closed interval from eight to nine is the closed interval from negative four to nine.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.