Video: Finding the Domain of the Sum of Two Functions

If 𝑓 and 𝑔 are two real functions where 𝑓(π‘₯) = π‘₯Β² βˆ’ 5π‘₯ and 𝑔(π‘₯) = √(π‘₯ + 1), find the domain of the function (𝑓 + 𝑔).

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Video Transcript

If 𝑓 and 𝑔 are two real functions where 𝑓 of π‘₯ is π‘₯ squared minus five π‘₯ and 𝑔 of π‘₯ is equal to the square root of π‘₯ 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 π‘₯. It’s π‘₯ squared minus five π‘₯. 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 π‘₯ is the set of real numbers.

And what about the function 𝑔 of π‘₯? 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 π‘₯, we have a function inside the square root, so π‘₯ plus one must be bigger than or equal to zero. And this means to find the domain of 𝑔 of π‘₯, we need to solve the inequality π‘₯ plus one is greater than or equal to zero. We’ll do this by subtracting one from both sides. And that tells us that π‘₯ must be greater than or equal to negative one. We can use interval notation to represent the domain of 𝑔 of π‘₯. π‘₯ must be greater than or equal to negative one, so we say that the domain of 𝑔 of π‘₯ 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 π‘₯ 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 π‘₯, is the domain of the function 𝑓 plus 𝑔.

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