Piecewise-Defined Functions

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Question

Piecewise-Defined Functions

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

b.

Accepted Answer

Welcome back everyone. It's another video. Write the piece wise formula of the given function. We're given the graph of a function which consists of three lines. So for this problem, let's recall that we can define a line as Y equals MX + B. What we're going to do for this problem is basically derive the equation of each line and then write the piece wise formula for the whole function. Let's begin with the first line. It is given to us in blue, and first of all, what we're going to do is identify the slope. Now slope is the change in y divided by the change in x. What we can do to simplify the problem is label the coordinates of the end points. So we're given the point at X equals -1, Y equals 3, and now for the other end point, the coordinates would be 1.1. So the change in Y would be Y2 minus Y1, so 1 minus 3. And the change in X would be X2 minus X1, 1 minus -1, and this gives us -2 divided by 2, which is -1. Now, let's identify the Y intercept B, which is Y minus MX, and we can once again choose any of the two points. So let's say we choose our second endpoint with Y of 1. And X of 1, so we subtract 1 minus -1 multiplied by 1, which gives us 2, and this is consistent with the given the intercept of Y equals 2. So the equation of this line is Y equals negative x + 2. And we can continue with the second line which is given to us in green. The end points have coordinates of 1.0 and 3.2. So in this case, our slope is 2 minus 0. Divided by 3 minus 1. Which gives us 2 divided by 2, and that's 1. For the Y intercept, we take Y minus MX, we can take the 0.1.0, this gives us 00 minus slope multiplied by the x value of 1. So 0 minus 1 gives us -1, meaning the equation of this line is X minus 1. Let's continue with the third line given to us in purple. The end points are 3.1. And 5.5. So the slope is 5 minus 1. Divided by the change in X, so 5 minus 3. This is 4 divided by 2, which gives us 2. And now the Y intercept B is Y minus MX, so we can take the 0.5.5, which gives us 5 minus 2 multiplied by 5. And that's -5. Now that we have those three equations, we can start writing the piece wise formula. We're adding the curly bracket, and we're saying that Y equals, let's begin with the first line. Which has an equation of negative X + 2, we then write a comma to specify the interval. So we have a closed circle at X equals -1 and a hollow circle at X equals 1. This means that X equals -1 is inclusive, so -1 is less than or equal to X, but 1 is not inclusive because we have an open circle. Well done. Now our second equation was X minus 1. Now the interval is from one inclusive because we have a closed circle. Up to 3 inclusive because once again our endpoint has a closed circle. And the final equation is Y equals 2 x minus 5. And now we want to specify the intervals. So from 3 not inclusive, we have an open circle. So 3 is less than X and X is less than or equal to 5 because the endpoint is a closed circle, and that will be our final answer to this problem. 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 Formulation

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