The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown...

Question

The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range.f(x) = cos⁻¹ x/2

Accepted Answer

Welcome back. I am so glad you're here. We're asked to perform transformation on the given graph of Y equals the inverse cosine of X to graph. The following function then identify the domain and range express the domain and range in interval notation. Our given function is F of X equals the inverse cosine of X divided by five. And we are given a graph, we have a vertical Y axis, a horizontal X axis and the graph is of Y equals the inverse cosine of X. The domain is from negative one to positive one inclusive of both. And the range is from zero to pi inclusive of both. And we can identify points of our graph. The farthest point to the left where an X value is negative one, our Y value is pi and then it curves to the next point which is at an X value of zero and A Y value of pi divided by two. And then that curves to the last point which is at an X value of positive one and A Y value of zero. So there's our Y equals the inverse cosine of X. But now we need to transform that to get and F of X equals the inverse cosine of X divided by five. Now, we really only see one difference in there from our original function. Now, we're just dividing that X by five. Another way to think about that is that we have the B value being multiplied by our X. What's being multiplied by our X? Here, it's 1/5. So B equals 1/5. Now, in order to transform this, there's a couple of ways of thinking about this first, we can see that something's happening directly to our X. Something's being multiplied by it. It's 1/5 and that 1/5 is in between zero and one. It's a fraction. And when that happens, when something is multiplied directly to our X, that tells us we're going to have something happen horizontally. And when that number is in between zero and one, it tells us there's going to be horizontal stretching. So because that B value is 1/5 we know there's going to be horizontal stretching. Now, specifically, what's the stretching going to be? What we do is we take each of our X values and divide them by B. So we're taking each X value and dividing it by 1/ or in other words, multiplying them all by five. So if our first point here is negative one pi and we are going to divide all the X values by 1/5 or we're multiplying it by one negative one, multiplied by five is negative five. So we'll have a new point at negative five pi we've got another point at zero pi divided by two while taking zero and multiplying that by five, that's still going to be zero. So this point will remain zero pi divided by two. And then that last point that we're given 10, we're going to divide that one by 1/5 or we're multiplying one by five, which will give us a positive five zero. And those three points can help us to graph and it will demonstrate the horizontal stretching. So let's move this to a different part of the screen where we've got a blank graph. And you can see that we now have a domain of what's shown on our graph. The horizontal X axis has a domain of negative six to positive six. And the Y values or the vertical axis for the Y values, our range is from negative three pi divided by two to positive three pi divided by two, we have our three points. And so the first one is at 50. So from the origin, we'll go to the right five and then up or down zero, then we have zero pi divided by two from the origin will go up to pi divided by two. And then our last point negative five pi from the origin will go to the left five and then up pi and then we want to try to connect these as smoothly as possible. And there is our F of X equals the inverse cosine of X divided by five. Now, we're also asked to list the domain and range in interval notation interval or the domain. Excuse me is all the possible X values. Well, that's going to start at. So we put a bracket negative five and that goes up to a positive five and close that with a bracket. And then the range, that's all the possible Y values that one is also inclusive. So we start with a bracket that's going to start at zero and still goes up to PI and we close that with a bracket and there our, there's our graph and our domain and our range well done. We'll catch you on the next one.

Key Concepts

Inverse Trigonometric Functions

Transformations of Functions

Domain and Range

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