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. Write the piece wise formula of the given function. We're given the graph of the piece wise function, and we can see that it consists of two different lines. We have to recall that in general a line can be described by Y equals MX plus B, where M is slope and B is the Y intercept. So let's begin with the orange line and define its equation. We're going to begin with the slope. M equals the change in Y divided by the change in X, and here we can take two points. So, first of all, we can take one of the Y coordinates, which would be 2. We're going to subtract the Y coordinate of the other point. We can take the very beginning where we have Y of 4, so we have 2 minus 4 divided by the corresponding change in X, right? So we're subtracting the X coordinate of the first point that we took, that's 1. I 0, right, because the starting point is 0.4, and now we can see that 2 minus 4 gives us -2 and -2 divided by 1 overall gives us -2 for the first slope. We also know that B is the Y intercept and our Y intercept is at 4. So overall, the first equation that we get is Y equals -2 X + 4, and now we need to define the domain, right? The sub interval where this line is present and basically what we can see is that the orange line is defined between 0 and 1, but we have to be very careful. First of all, we're going to write that X is between 0 and 1 not inclusive, and then we're going to pay attention to the circles at the end. Points we see that at the very beginning at x equals 0 we have a solid point and this means that 0 is included in the subinterval and now if we consider 1 we notice that we have a hollow or open circle, meaning 1 is not inclusive. Well then we have the first part of the piece wise formula, and then we're going to consider the second line, Y equals MX + B. We're going to identify the slope, change in Y divided by the change in X. We're going to take the end points. One of our points is 2.2, and the other one is 1.1. So to find the slope, we're taking the change in Y2 minus 1. Divided by the change in X to minus 1, and this gives us a slope of 1. Now we don't know the Y intercept. Of course we can extend the line and we can see that the intercept is the origin if this line was actually defined between 0 and 1, right? But in reality what we can do is just solve for the intercept using B equals Y minus MX, and we can take any point, let's say 1.1. So we are going to get 1 minus 1 multiplied by 1, which is 0. So our second line is Y equals 1 multiplied by X plus 0, which is just Y equals X. And then we can define the piece wise formula. We're going to introduce a curly bracket. We're going to begin with our first line -2 x + 4, and then we are including the subinterval. 0 is less than or equal to x and X is less than 1, and then our second line is Y equals x. And the sub interval is between 1 and 2 inclusive, right, because we have closed circles at the end points, so we can say that X is greater than or equal to 1 and less than or equal to 2, 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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