Graph each piecewise-defined function.

Question

f (x) = {\begin{matrix} 4 - x & if x < 2 \\ 1 + 2 x & if x \geq 2 \end{matrix}

Accepted Answer

Hey, everyone here we are asked to graph the following function F of X is equal to X plus five. If X is less than, or equal to three and 14 minus three, X, if X is greater than three, here we have four answer choice options. ABC and D, each of which display a graph representing our function. So the key to graphing our given function is to graph each interval of the domain separately. So we first want to consider just the first part of our function where we have F of X is equal to X plus five, which is defined for when X is less than or equal to three. So we want to begin by finding ordered pairs for this function. So we can start by substituting in X is equal to three into this function. So we have F of three is equal to three plus five which simplifies 28. So we have one ordered pair that we just found which is three and eight. Now we want to repeat this process for other values of X that are now less than three. So we can now choose the value, let's say have X be equal to one. So at X is equal to one, we have F of one which gives us one plus five, which equals six. And so now we have a second ordered pair of one and six repeating this process one more time, we can choose X is equal to two. So we have F of two is equal to two plus five, which equals seven. So we see a third ordered pair is at two and seven. So now with our three ordered pairs, the next step is to just plot these three points on a graph. In these three points, we have begun graphing this first part of our function. So now the next step is to just draw a line that connects these three points. And so doing this, we have a line that goes from the end point of three and eight all the way to negative infinity. Since we have X can be equal to three, we have a solid line for the end point since the function is defined at X is equal to three. However, X can also be less than three. So it goes all the way to negative infinity. So now want to repeat this process with the second part of our function. So considering the second interval F of X is equal to 14 minus three, X, if X is greater than three, we want to repeat this process by first finding ordered pairs. So we can begin when X is equal to three. So substituting in three into our function, we have F of three, which is equal to 14 minus three times three. Simplifying we get the output of five. So we see we have one ordered pair at three and five. Now again, repeating this process to find more ordered pairs, we will now choose values of X that are greater than three. So we can choose X is equal to four. So we have F of four is equal to 14 minus three times four. Simplifying we get two. So we have a second ordered pair at four and two and repeating this process one more time, we can choose X to be equal to five. So we have F of five is equal to 14 minus three times five. And simplifying we get negative one. So we see a third ordered pair is five and negative one. So now with these three ordered pairs for the second part of our original function, we now just want to plot these points on a graph. So plotting these points, we see we still have the line we drew from earlier. And now we have three new points that are underneath the line. So with these new points to finish up graphing the second part of our function, we need to draw a line connecting these points. And so drawing the line, we have our second part of our function here. And we see the end point is at three and five. And this endpoint is a hollow circle since we know that X or the function F of X is equal to 14 minus three, X is undefined when X is equal to three, since we know here that X must be greater than three. And so we go from three all the way to positive infinity since X can also or is equal to or is greater than three. So again, we go from three to positive infinity but three is excluded. So with these two lines drawn, we see we have our final graph here where we have the upper line representing the first part of our function. And the lower line is representing the second part of our function. And so with this final graph, we are left here with answer choice. C thanks for watching. I hope you found this video helpful and I'll see you again next time.

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

Key Concepts

Piecewise-Defined Functions

Graphing Piecewise Functions

Domain and Interval Notation

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