Video Transcript
Give the piecewise definition of the function 𝑔 whose graph is shown.
In this question, we’re given the graph of a function 𝑔. We need to use this to determine the piecewise definition of this function. And we can do this by recalling piecewise functions are defined by a number of subfunctions over different subdomains. And usually we can see the different subdomains of the function by looking at its graph. Normally, we can just look at the graph and check to see if it looks like several different graphs have been stitched together. For example, we can see on the left-hand side of this graph, we have a horizontal line. However, in the middle of this graph, it seems we have a parabola. And finally, on the right-hand side of this graph, we have a straight line. So it appears that 𝑔 is a piecewise-defined function with three subfunctions. Therefore, we can find the piecewise definition of our function 𝑔 by finding the equation of each of these subfunctions and determining the subdomain of each of these subfunctions.
Let’s start with the first subfunction. That’s the straight line on the left. We can see this is a horizontal straight line at 𝑦-coordinate one. And we know that a horizontal line at 𝑦-coordinate one is the straight line 𝑦 is equal to one. We now need to determine the subdomain of this function. We need to see which values of 𝑥 our function 𝑔 of 𝑥 outputs one. We can do this by considering the 𝑥-coordinates of points which lie on this horizontal line. Since there’s a filled-in dot at the endpoint of this horizontal line, we can see when 𝑥 is equal to one, our function 𝑔 outputs one. And we can also see that the same is true for any value of 𝑥 less than or equal to one. Therefore, 𝑔 of 𝑥 is equal to one if 𝑥 is less than or equal to one. This is our first subfunction and first subdomain. And we can add this to the piecewise definition of our function 𝑔 of 𝑥. 𝑔 of 𝑥 is equal to one if 𝑥 is less than or equal to one.
Let’s now move on to the second subfunction. That’s the parabolic section of the graph. Let’s start by finding the subdomain of this subfunction. We can see that this starts when 𝑥 is one. However, we do need to be careful since this section of the graph starts with a hollow dot. This means that 𝑥 is equal to one is not included in this subdomain. Let’s also determine the upper part of this subdomain. We can see that this is when 𝑥 is equal to four, since this is when the parabolic part of the curve ends. And it’s worth noting here we can include 𝑥 is equal to four into either subdomain, in the second subdomain or the third subdomain. We’ll choose to include it in the final subdomain. This gives us a subdomain of one less than 𝑥 less than four, or 𝑥 must be in the open interval from one to four.
We now need to determine the equation of this parabola. And we can do this by noting we’re given the two 𝑥-intercepts of the parabola. Therefore, we can write this parabola in factored form. It must be of the form 𝑘 multiplied by 𝑥 minus two times 𝑥 minus three. And of course we can also see in the diagram the parabola opens upwards. So the value of 𝑘 must be positive. And we can determine the value of 𝑘 by considering the coordinates of another point which lies on our curve. For example, we can see the point with coordinates four, two lies on this parabola.
Therefore, if we substitute 𝑥 is equal to four and 𝑦 is equal to two into the equation of the parabola, we must have the equation holds true. We must have that two is equal to 𝑘 times four minus two multiplied by four minus three. And we can solve this equation for 𝑘. Four minus two is equal to two, and four minus three is equal to one. So we get two is equal to two 𝑘, and we can divide through by two to see that 𝑘 is equal to one. Therefore, setting 𝑘 equal to one gives us the equation of the parabolic section of this diagram. And we know that this is the second subfunction of 𝑔 of 𝑥. 𝑔 of 𝑥 is equal to 𝑥 minus two multiplied by 𝑥 minus three if 𝑥 is greater than one and less than four.
Let’s now do this one more time for the third and final subfunction. Let’s start by finding the subdomain of this subfunction. Remember, we’re including 𝑥 is equal to four in this subdomain. And we can see the graph of this subfunction continues indefinitely. So all values of 𝑥 greater than or equal to four are included in this subdomain, which means the third subdomain of this subfunction is all values of 𝑥 greater than or equal to four. And we can write this as four being less than or equal to 𝑥.
We now need to find the equation of this straight line. There’s a few different ways of doing this. We’re going to use the point–slope form of the line. We do this by first noting the line passes through the point four, two, and we can find the slope of the line from the diagram. For every one unit we move to the right, the line moves one unit up. Therefore, its slope is one.
We can then recall the point–slope equation of a line tells us the equation of a line passing through the point with coordinates 𝑥 sub one, 𝑦 sub one and with a slope of 𝑚 is 𝑦 minus 𝑦 sub one is equal to 𝑚 times 𝑥 minus 𝑥 sub one. And we know that our line has a slope of one and passes through the point with coordinates four, two. Substituting these values into the point–slope equation of the line, we get 𝑦 minus two is equal to one times 𝑥 minus four. And finally, we simplify and rearrange this equation to get that 𝑦 is equal to 𝑥 minus two. So 𝑥 minus two must be the third subfunction of 𝑔 of 𝑥, and this is our final answer.
Therefore, we were able to show the piecewise definition of the function 𝑔 given in the diagram is 𝑔 of 𝑥 is equal to one if 𝑥 is less than or equal to one. 𝑔 of 𝑥 is equal to 𝑥 minus two times 𝑥 minus three if 𝑥 is greater than one and less than four. And 𝑔 of 𝑥 is equal to 𝑥 minus two if 𝑥 is greater than or equal to four.
