### Video Transcript

Find the domain of the
vector-valued function π of π‘ equals two π‘ squared π plus root π‘ minus one π
plus five over two π‘ plus four π.

Weβre looking to find the domain of
our vector-valued function. Now, each component function in our
vector-valued function will have its own domain, the set of values it can take. The domain of our vector-valued
function π will be the intersection of these three domains. So our job is to begin by
identifying the domain of each component function. Letβs begin by finding the domain
of the horizontal component two π‘ squared.

This is a polynomial. Now, we know that the domain of a
polynomial is simply the set of all real numbers. So the domain of this component
function is indeed the set of real numbers. And what about the domain of our π
component, the domain of the square root of π‘ minus one? Well, weβre interested in the
values the square root of π‘ minus one can take. And we know that the square root of
a negative number is not a real number. This means we need the expression
inside our square root π‘ minus one to be either equal to zero or greater than
zero. Solving for π‘ by adding one to
both sides and we obtain that π‘ must be greater than or equal to one. And we found the domain of this
component function.

Our final component function is
five over two π‘ plus four. We recall that the domain of the
quotient of two functions is equal to the intersection of the domain of each, but
where the denominator is not equal to zero. Our numerator and denominator are
polynomials. So the domain is all real
numbers. But we must ensure that two π‘ plus
four cannot be equal to zero. By subtracting four from both
sides, we see that two π‘ cannot be equal to negative four. And then dividing by two, we find
that π‘ cannot be equal to negative two.

So now, we have the domain of each
of our component functions. Remember, the domain of our
vector-valued function is the intersection of these. So itβs real numbers greater than
or equal to one, but not including negative two. Of course, negative two is less
than one. So our domain is simply real
numbers greater than or equal to one. And we can write this using
interval notation, as shown.