Video: Finding the Domain of a Vector-Valued Function

Find the domain of the vector-valued function π‘Ÿ(𝑑) = (𝑑² + 4) 𝑖 + tan (𝑑) 𝑗 + 2 ln (𝑑)π‘˜.

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Video Transcript

Find the domain of the vector-valued function π‘Ÿ of 𝑑 equals 𝑑 squared plus four 𝑖 plus tan 𝑑 𝑗 plus two times the natural log of 𝑑 π‘˜.

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 𝑑 squared plus four, the domain of tan 𝑑, and the domain of two times the natural log of 𝑑.

We’ll begin by looking at the function that represents the component for 𝑖. That’s 𝑑 squared plus four. 𝑑 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 𝑑 must be a real number. And we found the domain of 𝑑 squared plus four.

But what about the domain of tan of 𝑑? Let’s begin by recalling what the graph of 𝑦 equals tan of π‘₯ looks like. It looks a little something like this. Notice at πœ‹ by two and then intervals of πœ‹ radians, there is an asymptote. These are the values for which tan of π‘₯ is undefined. We can therefore say that the domain of tan 𝑑 is all values of 𝑑 not including 𝑑 is negative one πœ‹ by two, one πœ‹ by two, three πœ‹ by two, and so one. Another way of saying this is 𝑑 cannot be equal to two 𝑛 plus one multiples of πœ‹ by two, where 𝑛 takes an integer value. And now, we know the domain of tan of 𝑑.

Finally, let’s find the domain of two times the natural log of 𝑑. We know that the natural log of 𝑑 cannot take values of 𝑑 less than or equal to zero. So, the domain of the natural log of 𝑑, and therefore two times the natural log of 𝑑, is 𝑑 is greater than zero. Remember, the domain of our vector-valued function is the intersection of these domains. 𝑑 is therefore a real number greater than zero which is not equal to two 𝑛 plus one multiples of πœ‹ by two for integer values of 𝑛.

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