In Exercises 19–32, find the (a) domain and (b) range.

Question

𝔂 = 4 sin ( 1 )

x

Accepted Answer

Welcome back, everyone. Determine the domain and range for the function F of X equals 3 cosine of 2 divided by X. So let's begin with the domain of the function, and we have to recall that the domain is all of the X values for which our function is defined, right? So we're taking cosine of 2 divided by X, right? So if we generally take cosine of some value X, we can say that cosine is defined for any X. Well, in this case, we have to understand that we have 2 divided by X and division by 0 is undefined. So what we can do is simply say that 2 divided by X should meet the following condition X should not be equal to 0 because if X was 0, we would get division by 0, and this is undefined. So we have our domain, right? We can take any X value that we want except from 0, and we can divide 2 by any other number in real plane. So for our domain, we can say that X belongs to the interval from negative infinity. Up to 0 and from 0 up to positive infinity. In other words, all real numbers except from X equals 0. Now let's consider the range. And for range, those are just the Y values. That our function can get from minimum to maximum. Now let's recall that cosine is a periodic function which is Oscillating between -1 and 1 inclusive. What is the difference here now? The angle itself doesn't change the range. What actually changes the range is the amplitude. That's the number in front. So we want to incorporate that 3 into our inequality, and what we can do is simply multiply all of our sides by 3, right, which gives us -3 is less than or equal to 3 cosine of. 2 divided by x. Not that we're also changing the angle which has no effect on the range, which is less than or equal to 1 multiplied by 3, which is 3. This is how we have identified the range. Y belongs to the interval from -3, the minimum value up to 3, the maximum value. Both are inclusive, right? Well, once we have our answers now for the domain, X belongs to all real numbers except from 0, and for the range Y belongs to the interval from -3 inclusive up to 3 inclusive. Thank you for watching.

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = 4 sin ( 1 )
x

Key Concepts

Domain of a Function

Range of a Function

Trigonometric Functions

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