Graph each piecewise-defined function.

Question

f (x) = {\begin{matrix} - 2 x & if x < - 3 \\ 3 x - 1 & if - 3 \leq x \leq 2 \\ - 4 x & if x > 2 \end{matrix}

Accepted Answer

Hey, everyone here we are asked to graph the following function F of X is equal to three, X plus seven. If X is less than negative four and negative two, X plus two, if negative four is less than, or equal to X, which is less than, or equal to three and five, X minus 10. If X is greater than three, here we have four answer choice options. Each displaying a graph to represent each part of our given function. So to begin solving this problem, we want to graph each part of our given function, each domain of the function individually. So we can start with the first part of our function where we have F of X is equal to three X plus seven for when X is less than negative four. So to graph this function, we just want to consider arbitrary values of X that are within this inequality of, of X is less than negative four. So we can actually begin with X is equal to negative four. Since this will give us the end point of this function, although it is undefined at this point. So finding this value, we find F of negative four gives us three times negative four plus seven. Simplifying we get negative 12 plus seven. And now adding these values, we get negative five. So we see that one of our ordered pairs is at negative four and negative five. Now we want to repeat this same procedure to get other ordered pairs for our function that are less where the X values are less than negative four. So we can choose the value where X is negative six. So finding F of negative six, we get three times negative six plus seven. Now simplifying this expression, we get the value of negative 11. So we see another ordered pair we have is negative six and negative 11. Now we can also choose a value where X is equal to negative five. Now finding F of negative five, we have three times negative five plus seven. Now simplifying we get the value of negative eight. So we see a third ordered pair that we have here is negative five and negative eight. So now the next step is to start actually trying to graph this part of our function. So we can do this by first just graphing each of our ordered pairs. And so doing this, we see we end up with a graph with three of our points here, each individually graphed. And now the next step is to just connect these points through a partial line. And doing this, we see we have our line and that the end point at negative four and negative five is a hollow point since F of X is equal to three, X plus seven is not defined when X is equal to negative four. Since we need X to be less than negative four. Now, moving on, we want to consider the second part of our original function which is F of X is equal to negative two, X plus two if X is greater than or equal to negative four, while X is still less than or equal to negative or positive three. So now with this second part of our original function, we want to repeat a similar process at as we did earlier by considering arbitrary values of X. So we first want to consider the lowest value of X which is negative four. So when X is equal to negative four, we have F of negative four is equal to negative two times negative four plus two. And now simplifying, we see that we get the value of 10. So we see that one of our ordered pairs is negative four and 10. Now repeating this process with other values of X that are within the given interval, we can choose X is equal to zero since it is within negative four and three. So we have F of zero is equal to negative two times zero plus two. Simplifying we just get two. So we see a second ordered pair is zero. And two, repeating this process one more time, we can choose when X is equal to the highest possible value, which is three. So we have F of three is equal to negative two times three plus two. Simplifying we get the output of negative four. So we see a third ordered pair is at three and negative four. So now with our three different ordered pairs here, we now just want to plot the points as we did with the first part of our function. And so plotting, we see we still have, we are on the same graph as our first part of the function, but we now have the three additional points included for the second part of our function. And so just like earlier, we want to now connect these points with a line. And so connecting these points, we see that we have two end points and the end points are at negative four and 10 and three and negative four. And we see that these end points are solid circles since they are included within the interval. And now the last step here in this problem is to consider the final part of our original function, which is that N of X is equal to five, X minus 10 if X is greater than three. So now with this statement here, we just need to repeat the same process as earlier where we choose values of X to get ordered pairs and then plot the point and draw a line through them. In doing this, we see that we end up with an increasing line and the end point of this line is at three and five. And this is a hollow point since we know that X needs to be greater than three. And so it is undefined or this part of the function is undefined and X is equal to three. So with this, we have our three parts of our original function graph. So we have now three different lines. And with this finalized graph, we see that we are left with answer choice B. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Graph each piecewise-defined function.
$f (x) = {\begin{matrix} - 2 x & if x < - 3 \\ 3 x - 1 & if - 3 \leq x \leq 2 \\ - 4 x & if x > 2 \end{matrix}$

Key Concepts

Piecewise-Defined Functions

Graphing Linear Functions

Domain Restrictions and Continuity

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