### 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, weβre looking to find the
domain of this combination function. And so, we recall that the domain
of π plus π, thatβs the set of inputs that will yield real outputs, is the
intersections of the domains of π and π. So, letβs find the domains of π
and π. We begin with the function π of
π₯. Itβs π₯ squared minus five π₯. Itβs 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. Weβll 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 canβt really define
β, hence why we canβt 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.