In Exercises 37 and 38, write a piecewise formula for the func...

Question

In Exercises 37 and 38, write a piecewise formula for the function.

Accepted Answer

Welcome back everyone's another video. Find the piece wise formula of the graph below. For this problem, as we can see on the given coordinate plane, we have two separate lines, right? So we're going to identify the function in a form of Y equals curly bracket, and then we're going to fill out the two functions. Specifically, we want to identify the equation of each line. So let's begin with the orange one. First, let's real that a line is defined as Y equals MX plus B, where M is the slope and B is the Y intercept. So we can essentially begin with the orange line and identify the slope. To recall that slope equals rise divided by a run, so deltay divided by delta X, we can see that it's negative because with an increase in X, the value of Y decreases. So we're going to add a negative sign, and now our rise is -3. Our run is 3, right? So we can say -3 divided by 3 gives us -1. And now we can clearly see that the intercept. B is equal to 3, so we can conclude that for the orange line Y equals MX plus B, so that's negative X plus 3. And now in the piece wise formula we want to identify the X value, so basically the subinterval for the domain, right? We can notice that the orange line extends from -1 up to 2, right? And it's really important to understand that X equals -1 is within the domain because we have a solid circle. So we're going to say that X is between -1 inclusive and 2 inclusive. That's because both of these circles are solid circles, or basically fully filled, right? So we're going to say that Y equals negative X plus 3 if X is between -1 and 2 inclusive. And then we have another line, right? So, for the blue line, let's identify the slope, change in Y divided by the change in X. So we can see that we're going down to units and we're going to the right one unit, right? So that would be -2 divided by 1, which is -2. So we have our slope. And now since we don't have any intercept, we can just use one of the points, for example, 3.3 and solve 4B. Using our equation B equals Y minus MX. So if we take 33, we get. -3 for Y minus MX, so M is -2, where subtracting -2 multiplied by the X value, which is 3, and -3 minus -2 multiplied by 3 gives us -3 + 6, which is basically 3, right? So our next equation is Y equals. 2 X plus 3, if X is between 2, not inclusive because we have a hollow circle, so X should be greater than 2, but less than or equal to 3, right? Because we once again have a close circle. So our second part of the piece wise formula would be -2 X plus 3. If X is greater than 2, but less than or equal to 3, and those will be our final answer. So let's label our formula and thank you for watching.

In Exercises 37 and 38, write a piecewise formula for the function.

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Key Concepts

Piecewise Functions

Domain and Range

Function Notation

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