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. Determine the piece wise formula of the given function. Our graph shows that the given function consists of two lines. So what we're going to do is basically recall that a line has a form of Y equals MX plus B, where M is the slope, and B is the Y intercept. So we're going to define the equation of each line starting with the blue line. We need to take the endpoints of the line. The first endpoint is at X equals -1, and Y of -1, and the second endpoint is at X equals 0, Y equals 3. So we can determine the slope, which is defined as the change in Y divided by the change in X. We can take Y2 minus Y1, that's 3 minus -1, and divide that by the change in X 0 minus -1. Which gives us 4 divided by 1 and that's 4. To identify the Y intercept, we can solve for B, which gives us Y minus MX, and we can take any point that we wish, for example, 0.3, which gives us 3 minus 4 multiplied by 0, and that's 3, which is consistent with the given Y intercept. So we can say that the equation of this line is Y equals MX, so 4 X. Plus B Plus 3. Now we can continue with the second line given to us in green. The endpoints are the origin 0.0 and 1.2. So the slope is 2 minus 0. Divided by 1 minus 0. Which is to The Y intercept is Y minus MX. We can take the origin and say that this is 0 minus 2 multiplied by 0 and that's 0. So the equation of the blue line. Is 2X + 0 or simply 2X. And now we can write the piece wise formula. We are adding Y equals curly bracket. We begin with the equation of the first line. Or X + 3, and then we add a comma to specify the interval. So X is between -1 and 0. And here we have to be careful. We have to notice that the 1st and 0.-1 has a closed circle, meaning -1 is inclusive, so -1 is less than or equal to X and X is less than 0 strictly since we have an open circle at a 0.0.3. Now, we are including the equation of the second line to X, we write comma. And include the interval. X is between 0 and 1. And we noticed that both endpoints are closed circles, meaning those two points are inclusive, so X is between 0 inclusive and 1 inclusive. And that will be our final piece wise formula. Thank you for watching.

Piecewise-Defined Functions

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

a. <IMAGE>

Key Concepts

Piecewise-Defined Functions

Graph Interpretation

Function Formulation

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