# Question Video: Finding the Domain and Range of a Trigonometric Function When Given its Graph Mathematics

The following graph shows the function 𝑓(𝜃). Assume the function has a period of 2𝜋. (i) What is the domain of 𝑓(𝜃)? (ii) What is the range of 𝑓(𝜃)?

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### 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 know that all the characteristics of a periodic function are contained over an interval of this length. In this question, we are told the period is equal to two 𝜋. Therefore, we only need to consider the graph between zero and two 𝜋. The domain of any function is a set of all possible input values. And we can see from the graph that the function is well defined at all values of 𝜃. We can therefore conclude that the domain of 𝑓 of 𝜃 is all real numbers written as the open interval from negative ∞ to ∞.

The range of any function is the set of all output values. From the graph, we see that the function oscillates and is continuous between negative seven and three. The maximum value of the graph is three, and the minimum value is negative seven. We can therefore conclude that the range of 𝑓 of 𝜃 is the set of values on the closed interval from negative seven to three. The two answers to this question are the open interval from negative ∞ to ∞ and the closed interval from negative seven to three.

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