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

Question

𝔂 = tan(2x - π)

Accepted Answer

Welcome back, everyone. Find the domain and range of the function G of X equals cotangent of X minus pi. First of all, we are going to analyze G of X and let's recall that cotangent is simply the ratio between cosine and sine. So we can express GFX as cosine of x minus pi divided by sin. Of x minus y. And now we can now see that this is a rational function which is undefined whenever its denominator is equal to 0, meaning for the domain we're going to say that sign of x minus i should be not equal to 0. And then when is sine equal to 0? Well, sine is equal to 0 when it has an angle that is a multiple of pi. So we're going to take our angle, which is X minus pi, and we're going to say that this angle cannot be an integer multiple of pi. So let's assume that our integer is M and we're multiplying that integer by pi. Notice that M. Is an integer. So what we're going to do is simply rewrite this expression as x is not equal to mpi. Plus pie, so we're just adding pie to both sides and now we can just Factor out i, and we can say that x is not equal to pi in prism + 1. Now simply speaking, we can say that N + 1 is another integer k, right, because we have an infinite set of integers, so there's no necessity to write x is not equal to n + 1 pi. We can just say that x is not equal to. I multiplied by K. Right, so this just looks simpler and more accurate. We can just say that K is some new integer, which is simply shifted, but since we have an infinite set of integers, it doesn't really matter if we have N+1 or K. It basically represents the same integers, the same infinite set of integers. So we have defined our domain. Now for range, we simply have to recall. The graph of a cotangent and we have to understand that it is a periodic function if we just consider one of the periods, well, we are going to observe a function that is decreasing, right? So essentially it goes from positive infinity. Then goes to negative infinity. So for our range, the cotangent function can take any value between negative infinity and positive infinity. And that said, we can essentially conclude our findings. We can say that the domain. Of the given function. Is X not equal to ik where k belongs to integer C. And now for our range. The Yal Belongs to all real numbers from negative infinity up to positive infinity and those will be our final answers to this problem. Thank you for watching.

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

𝔂 = tan(2x - π)

Key Concepts

Domain of a Function

Range of a Function

Transformation of Functions

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