In Exercises 31–32, the domain of each piecewise function is (-∞,...

Question

In Exercises 31–32, the domain of each piecewise function is (-∞, ∞)(a) Graph each function.(b) Use the graph to determine the function's range.

Accepted Answer

Hello today, we're going to go ahead and find the range of a piecewise function. So let's go ahead and look at the different conditions of this piecewise function. The first condition states that f of X is equal to X squared if X is greater than one. So let's go ahead and identify what that part of the piecewise function is. So just to go ahead and write it right here, F of X is equal to X squared and we only want the part where X is greater than one, so F of X is equal to X squared. Is the graph of a positive parabola, where the parabola starts at the origin and we have the points one comma one and negative one comma one. Now again this condition here states that we only want the part of this graph where X is greater than one. So we want to go ahead and include only this part of this parabola. And of course for this specific part, we're not gonna include the value of one because this statement says that we only want values that are greater than one, not including one itself. Now, let's go ahead and identify the second part of our piecewise function. The second part of our piecewise function states that we have F of X is equal to X only when X is less than or equal to one. So just to go ahead and write that right here, F of X is equal to X and we only want the part where X is less than or equal to one. Well f of X is equal to X is the graph of a positive line. So we're gonna have this line right here And this line starts at the origin 0, 0. We have the points negative one comma negative one. And we have the point here positive one, positive one. Now, of course this condition states that we only want the values of X that are less than one. Also including one. So we're only gonna want this portion of this linear graph for a piecewise function and get to go ahead and include one because our condition says that we want the value of one here. So all we need to do now is just go ahead and combine everything together on the same axis for our piecewise function. So putting everything together, we know that the common point between these two graphs is going to be one comma one. So let's just go ahead and put that right here, One comma one. And because the piecewise function is a combination of multiple functions, we can go ahead and include that point on our piecewise function. So for X greater than one, we only want the shape of a positive parabola. So we're gonna have this kind of shape for X greater than one. And for X less than one we're going to have a negative line or a positive line but going in the negative direction and passing through the origin 00. So this is going to be the shape of the other half of our piecewise function. And that's going to be the graph of our piecewise function. Now, what we need to do is to go ahead and identify the range. Well recall that the range is the lowest possible Y value to the highest possible Y value of our graph. When we're looking at this piece of the piecewise function, as X continues to decrease, our Y values are going to decrease infinitely. That means that the y values decreasing are gonna be approaching negative infinity. And if we look at this portion of the piecewise function as our x values continue to increase, our Y values are gonna continue to increase infinitely. So the highest possible Y value is going to be positive infinity. That means that the range of this graph is going to be from negative infinity to positive infinity. And that's gonna be our solution. That also tells us that the solution to this problem is going to be C. So, I hope this video helps you in identifying the range of a piecewise function. And I'll go ahead and see you all in the next video

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In Exercises 31–32, the domain of each piecewise function is (-∞, ∞)(a) Graph each function.(b) Use the graph to determine the function's range.

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