Question Video: Finding the Domain and Range of a Trigonometric Function from Its Graph Mathematics

Video Transcript

The following graph shows the function 𝑓 of 𝜃. Assume the function has a period of two 𝜋. What is the domain of 𝑓 of 𝜃? What is the range of 𝑓 of 𝜃?

We are told that this function has a period of two 𝜋. Now, we can recall that all the characteristics of a periodic function are contained in the graph of that function over an interval whose length is equal to the period. This function has a period of two 𝜋. So, we only need to consider its graph over an interval of length two 𝜋. We can use any interval of length two 𝜋 that we choose. So, let’s use the interval from zero to two 𝜋. The domain of a function 𝑓 of 𝜃 is the set of all possible values of 𝜃 such that the function is well defined.

In particular, given a graph of a function, its domain is the part of the horizontal axis where the graph exists. From the graph, we can see that this function is defined at all values of 𝜃 between zero and two 𝜋. And by extension, it’s therefore well defined for any real value of 𝜃. The domain of 𝑓 of 𝜃 is therefore the set of all real numbers. Or we can write this as the open interval from negative ∞ to ∞.

Next, we consider the range of 𝑓 of 𝜃. The range of a function is the set of all possible values of the function itself. Given the functions domain from a graph of a function. The range is the part of the vertical axis where the graph exists. Again, we need only look in the interval from zero to two 𝜋. And in doing so, we see that the minimum value of the function in this interval is negative seven and the maximum value is positive three. 𝑓 of 𝜃 is a continuous function. And so, it takes all values between its minimum and maximum. The range of 𝑓 of 𝜃 is therefore the closed interval from negative seven to three.

So, by considering the graph of 𝑓 of 𝜃 on the interval from zero to two 𝜋, which is an interval whose length is equal to the period of the function, we’ve found that the domain of 𝑓 of 𝜃 is the set of all real numbers, or the open interval from negative ∞ to ∞, and the range of 𝑓 of 𝜃 is the closed interval from negative seven to three.