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

Video Transcript

Consider the following graph of 𝑓 of 𝜃. What is the domain of 𝑓 of 𝜃? What is the range of 𝑓 of 𝜃?

Before we answer these questions about the domain and range of the function 𝑓 of 𝜃, let’s have a look at the graph. Now we note that the graph of this function repeats indefinitely. We see the same pattern over and over. And so 𝑓 of 𝜃 is a periodic function. We can see that the function has a local minimum when 𝜃 is equal to zero and the function returns to the same value when 𝜃 is equal to two 𝜋 radians. The function keeps returning to this local minimum at every integer multiple of two 𝜋. And so the period of this function is two 𝜋.

Let’s now consider the domain of this function. We recall that the domain of a function is the set of all possible input values such that the function is defined. In particular, on the graph of a function, the domain is the part of the horizontal axis where the graph exists. Now, we can recall that if a periodic function is defined within an interval whose length is equal to the period of that function, then the function is defined for all real numbers. From the graph, we can see that 𝑓 of 𝜃 is defined for all values within the closed interval from zero to two 𝜋, and the length of this interval is equal to the period of two 𝜋. So if the function is defined on this interval, it will be defined on the entire set of real numbers, which we can express as the open interval from negative ∞ to ∞.

Next, let’s consider the range of 𝑓 of 𝜃. We recall that the range of a function is the set of all possible values the function itself can take given the function’s domain. In particular, given the graph of a function, its range is the part of the vertical axis where the graph exists. Now, as we’ve already said, this function is periodic with a period of two 𝜋. We can recall that if a function is defined for all real numbers, which we found this function is, then its range over all the real numbers is equal to its range over an interval whose length is equal to the period. So we just need to consider the range of this function over an interval of length two 𝜋. Let’s choose the closed interval from zero to two 𝜋.

From the graph, we can see that the minimum value of the function over this interval is negative four and the maximum value is positive six. 𝑓 of 𝜃 is a continuous function. And so it takes all values between its minimum and maximum. Therefore, the range of the function 𝑓 of 𝜃 is the closed interval from negative four to six. We found then that the domain of the function 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 four to six.