Find the domain of the
vector-valued function 𝑟 of 𝑡 equals 𝑡 squared plus four 𝑖 plus tan 𝑡 𝑗 plus
two times the natural log of 𝑡 𝑘.
We recall that the domain of a
function is all the values it can take, all the possible inputs. And we can find the domain of a
vector-valued function by looking for the intersection of the domains of each of our
component functions. So, we’re going to need to work out
the domain of 𝑡 squared plus four, the domain of tan 𝑡, and the domain of two
times the natural log of 𝑡.
We’ll begin by looking at the
function that represents the component for 𝑖. That’s 𝑡 squared plus four. 𝑡 squared plus four is a
polynomial function. And we know that the domain of a
polynomial function is all real numbers. So, we can say that 𝑡 must be a
real number. And we found the domain of 𝑡
squared plus four.
But what about the domain of tan of
𝑡? Let’s begin by recalling what the
graph of 𝑦 equals tan of 𝑥 looks like. It looks a little something like
this. Notice at 𝜋 by two and then
intervals of 𝜋 radians, there is an asymptote. These are the values for which tan
of 𝑥 is undefined. We can therefore say that the
domain of tan 𝑡 is all values of 𝑡 not including 𝑡 is negative one 𝜋 by two, one
𝜋 by two, three 𝜋 by two, and so one. Another way of saying this is 𝑡
cannot be equal to two 𝑛 plus one multiples of 𝜋 by two, where 𝑛 takes an integer
value. And now, we know the domain of tan
Finally, let’s find the domain of
two times the natural log of 𝑡. We know that the natural log of 𝑡
cannot take values of 𝑡 less than or equal to zero. So, the domain of the natural log
of 𝑡, and therefore two times the natural log of 𝑡, is 𝑡 is greater than
zero. Remember, the domain of our
vector-valued function is the intersection of these domains. 𝑡 is therefore a real number
greater than zero which is not equal to two 𝑛 plus one multiples of 𝜋 by two for
integer values of 𝑛.