Graphing functions Sketch a graph of each function.

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Question

Graphing functions Sketch a graph of each function.

ƒ(x) = { 2x if x ≤ 1 , 3-x if x > 1

Accepted Answer

Welcome back everyone to another video. Consider the piece wise function H of X is equal to 3 x + 1 if x is less than or equals to 2 and 2 x + 3 if X is greater than 2, plus the graph of the function. Whenever we're given a piecewise function, the first point of interest in this case would be X equals 2 because it divides the function into two parts. The first part is H of X is equal to 3 x + 1 if X is less than or equal to 2, and we have to recognize that this is an equation of a line. So we want to plot a line and before we do that, we are going to draw a vertical line passing through X equals 2 because this line divides the piece wise function into two parts. So we want to plot the line on the left side. And if we want to plot a line, well, essentially we need to take two points, and we can take any X values that we want as long as they are less than or equal to 2. So the first one that we're going to take is X equals 0 and the other one is X equals 2. Now if we have those two points, we can connect them in a straight line, extend the line, and we are done. So we're going to evaluate the Y values starting with H of 0. And then H of 2. So H of 0 is going to be equal to 3 multiplied by 0 + 1, which is 1, and each of 2 is 3 multiplied by 2 + 1, which is 7. So now we have 2 points. The first one is 0.1. The second one is 2.7. What we want to do is locate those points, so we have 0. For X and 1 for Y. So let's add that point and then our next point is 2.7. So we have 56, and 7, right? So now we have those two points and what we can do is simply connect them in a straight line. Let's go ahead and do that. And now from here we can simply extend our line. To the left We don't want to cross X equals 2, right, because essentially on the right side we're going to have a different function. Now we can remove that line and continue with the second function. Before we draw the second function, it is really important to understand that the left part is defined at X equals 2. So we want to draw a closed or solid circle at the end, right, clearly indicating that this part is defined for the left curve. And now for the right one we're given H of X is equal to. 2 X + 3 if X is greater than 2. And what we're going to do here is also understand that this is a straight line. First of all, we're going to evaluate the limit. Of 2 X + 3. S X approaches to. From the right. And we can see this by direct substitution, right? That's. 2 multiplied by 2 + 3. Which is equal to 7 as well, right? So essentially this means that our function is continuous at 0.2 because it has the same value of 7. So we have one of our points, and now for the second point, let's suppose that X is equal to 6. We need an additional point, and then H at 0.6 is going to be equal to 2 multiplied by 6 plus 3, which is 15. So what we want to do is locate X equals 6, Y equals 15, and we simply want to plot the second part of the piece wise function. Notice that because the function is continuous, we don't actually need to include a solid circle at X equals 2, right? We can essentially draw another line. Passing through x equals 2. And X equals 6, right, for Y values of 7 and 15 respectively. And that's it. We have our piece wise function. We want to make sure that it. Passes through. X equals. 6, so let's simply remove that. Additional point And also let's remove the solid circle. Well done. So now let's draw the other line with a Different or the same color, we can use the same color, so we want to make sure that it passes through 6. 15. So when we have X equals. 6 Y is equal to 15. And that's it. We can simply extend that line. To the right And that will be our final answer to this problem. Thank you for watching.

Graphing functions Sketch a graph of each function.

ƒ(x) = { 2x if x ≤ 1 , 3-x if x > 1

Key Concepts

Piecewise Functions

Graphing Techniques

Continuity and Discontinuity

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