Question Video: Finding the Domain of a Given Sine Function Mathematics

Find the domain of the function 𝑓(𝜃) = 5 sin (𝜃) + 4.

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Video Transcript

Find the domain of the function 𝑓 of 𝜃 equals five sin 𝜃 plus four.

We recall first that, in general, the domain of a function 𝑓 of 𝜃 is the set of all possible input values 𝜃 such that the function is defined. The function we’re interested in here is 𝑓 of 𝜃 is equal to five sin 𝜃 plus four. We recall that the domain of the sine function itself is the set of all real numbers, which we can write as the open interval from negative ∞ to ∞.

In order to answer this question, we need to consider the functional transformations that have been applied to sin 𝜃 to give five sin 𝜃 plus four. Multiplying sin 𝜃 by five is a vertical stretch of the function by a scale factor of five. Adding four to the function is a vertical shift four units up, or we may say by four units in the positive 𝑦-direction. Each of these transformations has a vertical effect. And so they each affect the output values or range of the sine function. There’s been no change to the input of the function, and so the domain is unaffected.

The domain of the function 𝑓 of 𝜃 is, therefore, the same as the domain of the sine function, which is the open interval from negative ∞ to positive ∞.