Graph each piecewise-defined function.

Question

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

Accepted Answer

Hey, everyone here we are asked to graph the following function F of X is equal to X plus four. If X is less than, or equal to one and five, if X is greater than one here, we have four answer choice options. ABC and D, each of which display a graph representing our given function. So to graph our given function, we just need to graph each interval of the domain separately. So we can start with the first part of our function where we have F of X is equal to X plus four if X is less than or equal to one. So with this function here we want to substitute in values of X so that we can obtain ordered pairs for this function. So we can start with the highest value of X according to our inequality which is where X is equal to one. So now substituting in one into our equation, we have F of one is equal to one plus four which simplifies to five. So we see our first ordered pair here is one and five. So now we can repeat this process for other values of X that are now less than one. So we can repeat for, let's say X is equal to zero. So we have F of zero is equal to zero plus four, which simplifies to four. So we see that a second ordered pair, we have F now is zero and four. And now repeating this one more time, we can have X P equal to negative one. So we have F of negative one is equal to negative one plus four, which simplifies 23. So we see a third ordered pair is negative one and three. So now with our three ordered pairs, our next step is to plot these points on a graph. So plotting these points, we see we have our three individual points plotted on the same graph. And so we have begun graphing the first part of our function. And so the next step is to draw a line that passes through each of these points and that matches the interval where X is less than or equal to one. So drawing in this line that fits within the interval, we have an end point at one and five which is a solid end point. So since we know that F of X is equal to X plus four is defined for when X is equal to one, so we go from one all the way till negative infinity because X can also be less than one. So again, we go from one which is included all the way to ne negative infinity. And so we have graphed the first part of our function, which means we now need to consider the second interval of our function where F of X is equal to five, which is defined for when X is greater than one. So now looking at this equation here, we have F of X is equal to five, which is the, defined as Y is equal to five. So we know here that our second interval will be a horizontal line at Y is equal to five and it will go from one, X is equal to one all the way until positive infinity since X is greater than one. So with all this information, we can go ahead and graph this second part, this second interval of our given function. And in doing this, we now have a graph containing both parts of our function. And we see that this orange horizontal line is the second interval where Y is equal to five or F of X is equal is equal to five. And we see we go from X is equal to one until positive infinity. But it is worth noting that we know that F of X is equal, equal to five is undefined and X is equal to one since we know X must be greater than one for this part of the function. So with our final graph here we are left with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

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

Key Concepts

Piecewise-Defined Functions

Graphing Functions with Domain Restrictions

Continuity and Discontinuity in Piecewise Functions

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