Question Video: Finding the Domain of the Sum of Two Functions | Nagwa Question Video: Finding the Domain of the Sum of Two Functions | Nagwa

# Question Video: Finding the Domain of the Sum of Two Functions Mathematics • Second Year of Secondary School

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If π and π are two real functions where π(π₯) = π₯Β² β 5π₯ and π(π₯) = β(π₯ + 1), find the domain of the function (π + π).

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### Video Transcript

If π and π are two real functions where π of π₯ is π₯ squared minus five π₯ and π of π₯ is equal to the square root of π₯ plus one, find the domain of the function π plus π.

Firstly, we recall that the combination function π plus π is simply the sum of the functions π and π. Now, weβre looking to find the domain of this combination function. And so, we recall that the domain of π plus π, thatβs the set of inputs that will yield real outputs, is the intersections of the domains of π and π. So, letβs find the domains of π and π. We begin with the function π of π₯. Itβs π₯ squared minus five π₯. Itβs simply a polynomial, and we know that the domain of a polynomial function is the set of real numbers. So, the domain of π of π₯ is the set of real numbers.

And what about the function π 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 π 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 π of π₯, we need to solve the inequality π₯ plus one is greater than or equal to zero. Weβ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 π of π₯. π₯ must be greater than or equal to negative one, so we say that the domain of π of π₯ is the left-closed, right-open interval from negative one to β.

Notice that we canβt really define β, hence why we canβt have a square bracket on the right-hand side of this interval. And so we know that the domain of our function π plus π is the intersection, the overlap, between these two domains. If we consider the domain of π of π₯ to be a subset of the set of real numbers, then we see that the domain of π plus π, the overlap, is in fact the left-closed, right-open interval from negative one to β. And so, that interval, that set of values for π₯, is the domain of the function π plus π.

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