Video: Finding the Domain of the Sum of Two Functions

Video Transcript

If 𝑓 and 𝑔 are two real functions where 𝑓 of 𝑥 is 𝑥 squared minus five 𝑥 and 𝑔 of 𝑥 is equal to the square root of 𝑥 plus one, find the domain of the function 𝑓 plus 𝑔.

Firstly, we recall that the combination function 𝑓 plus 𝑔 is simply the sum of the functions 𝑓 and 𝑔. Now, we’re looking to find the domain of this combination function. And so, we recall that the domain of 𝑓 plus 𝑔, that’s the set of inputs that will yield real outputs, is the intersections of the domains of 𝑓 and 𝑔. So, let’s find the domains of 𝑓 and 𝑔. We begin with the function 𝑓 of 𝑥. It’s 𝑥 squared minus five 𝑥. It’s simply a polynomial, and we know that the domain of a polynomial function is the set of real numbers. So, the domain of 𝑓 of 𝑥 is the set of real numbers.

And what about the function 𝑔 of 𝑥? Well, with a root function, we know that to yield a real output, the number inside the square root must be bigger than or equal to zero. In 𝑔 of 𝑥, we have a function inside the square root, so 𝑥 plus one must be bigger than or equal to zero. And this means to find the domain of 𝑔 of 𝑥, we need to solve the inequality 𝑥 plus one is greater than or equal to zero. We’ll do this by subtracting one from both sides. And that tells us that 𝑥 must be greater than or equal to negative one. We can use interval notation to represent the domain of 𝑔 of 𝑥. 𝑥 must be greater than or equal to negative one, so we say that the domain of 𝑔 of 𝑥 is the left-closed, right-open interval from negative one to ∞.

Notice that we can’t really define ∞, hence why we can’t have a square bracket on the right-hand side of this interval. And so we know that the domain of our function 𝑓 plus 𝑔 is the intersection, the overlap, between these two domains. If we consider the domain of 𝑔 of 𝑥 to be a subset of the set of real numbers, then we see that the domain of 𝑓 plus 𝑔, the overlap, is in fact the left-closed, right-open interval from negative one to ∞. And so, that interval, that set of values for 𝑥, is the domain of the function 𝑓 plus 𝑔.