Piecewise-Defined Functions

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Question

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

a.

Accepted Answer

Welcome back, everyone. Write a piecewise formula for the given function. We're given the graph of the function, and we can see that it consists of three lines. One of them is orange, the other one is blue, and the third one is green. So let's remember that the slope intercept form for any line is Y equals MX plus B. We want to identify the slope and the intercept for each line. Let's begin with the orange. We're going to call it O. And basically we're going to identify the slope. Let's recall that we're taking Delta Y divided by delta X, and this means that we need two points. One of the points that we have is 0.3, and the endpoint is. 1,2, right? So we're going to use those points in the evaluation of the slope. Let's begin with Y2, which is 2, subtract Y1, which is 3, and divide by X2 minus X1, so 1 minus 0. This gives us 2 minus 3 divided by 1, which is -1. So we have our slope, and we can visually see that the intercept is at 3, so our B is 3, right? And we can say that the equation of the orange line is Y equals negative X plus 3. So we're going to label it as equation 1 and just move on to The second line, which is the blue one, so let's call it B. Let's not that it is a horizontal line. Now, what does that mean? Well, it has a slope of 0, so all that we have to do is basically identify the intercept. And if we extend that line, we can see that the intercept is Y equals 2. So the second line is simply Y equals 2. Let's label it as equation number 2, and then let's move on to the green line. Let's identify the slope. Change in Y divided by change in X, we need two points once again. The first point that we can use is X equals 2 Y equals 0, and the other end point is X equals 4, Y equals 2. So the slope becomes 2 minus 0. Divided by 4 minus 2. That's 2 divided by 2, and this gives us a slope of 1. Now we don't have the Y intercept, so what we can do is simply solve for the intercept using B equals. Y minus MX, right, we're basically rearranging the equation of the line and we can take any point that we want. Let's say 2.0. This gives us 0 minus 1 multiplied by X, which is 2. So 0 minus 2 gives us negative 2, and overall this line is Y equals x minus 2. That's equation number 3. And now what we want to do is basically write the piece wise formula Y equals curly bracket, and now we need to Write out the equation of each line starting with the orange one. We know that it is negative X + 3. We then write comma and we need to specify the subdomain. So basically, we can see that. X equals 0 is included, right, because we have a closed circle. So we're starting with 0, which is less than or equal to X and X is less than 1 strictly, right, because we have an open circle, which essentially means that X equals 1 is not inclusive. So this is how we get our first part, for the second part we see that Y equals 2. And now we need to specify the subdomain. Once again we can see that X is greater than 1, strictly greater, right, because we have an open circle, but less than or equal to 2, right, because we have a closed circle, so that 1 is less than X and X is less than or equal to 2. And finally, our third equation is X minus 2, specifying the domain. We have 2 not inclusive because there's an open circle is less than X and X is less than or equal to because once again we have a closed circle. 4 And that's it. We have our final expression for the piece wise function. Thank you for watching.

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

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Key Concepts

Piecewise-Defined Functions

Graph Interpretation

Function Continuity and Discontinuity

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