Question Video: Determining the Range of Piecewise-Defined Functions Mathematics

Find the range of the function ๐‘“(๐‘ฅ) = 8๐‘ฅ, if ๐‘ฅ โˆˆ [0, 1), and ๐‘“(๐‘ฅ) = 8, if ๐‘ฅ โˆˆ [1, 7], and ๐‘“(๐‘ฅ) = 15 โˆ’ ๐‘ฅ, if ๐‘ฅ โˆˆ (7,15].

06:20

Video Transcript

Find the range of the function ๐‘“ of ๐‘ฅ is equal to eight ๐‘ฅ if ๐‘ฅ is in the left-closed, right-open interval from zero to one and ๐‘“ of ๐‘ฅ is equal to eight if ๐‘ฅ is in the closed interval from one to seven and ๐‘“ of ๐‘ฅ is equal to 15 minus ๐‘ฅ if ๐‘ฅ is in the left-open, right-closed interval from seven to 15.

In this question, weโ€™re asked to find the range of a given piecewise function. And we can start by recalling the range of any function is the set of all output values of the function given its domain or the set of input values. And thereโ€™s many different ways of finding the range of a function. Since weโ€™re given a piecewise-defined function where each of the three subfunctions are linear, weโ€™ll do this by sketching its graph.

Before we sketch our graph, letโ€™s determine the domain of this function. Thatโ€™s the set of possible input values for our function. To do this, we know we have a piecewise-defined function. And the domain of any piecewise-defined function is the union of its subdomains. In other words, we can only input values of ๐‘ฅ into our function which are in these three sets. So weโ€™ll start by sketching our coordinate axes, where on the ๐‘ฅ-axis, we need to mark the endpoints of our subdomains. Thatโ€™s zero, one, seven, and 15. And it is worth noting we may need to extend this to include negative values of ๐‘ฆ. However, weโ€™ll see in this case itโ€™s not necessary.

We now need to sketch each subfunction separately over its subdomain. Letโ€™s start with the first subfunction defined over the left-closed, right-open interval from zero to one. We can see that this function is the linear function eight ๐‘ฅ. Since this is a linear function defined over an interval, this will be a line segment. And the easiest way to sketch a line segment is to find the coordinates of its two endpoints. To find the endpoints of this line segment, we need to substitute the endpoints of our subdomain into the subfunction.

Letโ€™s start by substituting ๐‘ฅ is equal to zero into our subfunction. We get eight multiplied by zero, which is equal to zero. Since zero is in the subdomain of this function, this tells us that ๐‘“ evaluated at zero is equal to zero, which in turn tells us the graph of our function passes through the origin. Weโ€™ll mark this with a solid dot. We now want to check the other endpoint of our subdomain. However, we do need to notice that this side of our interval is open. This means we canโ€™t evaluate ๐‘“ at one by substituting it into the subfunction eight ๐‘ฅ. However, we can use this to find the other endpoint of our subfunction.

Substituting ๐‘ฅ is equal to one into the subfunction eight ๐‘ฅ, we get eight multiplied by one, which is equal to eight. This is then the ๐‘ฆ-coordinate of the endpoint of our first subfunction. The endpoint of our subfunction will be one, eight. So weโ€™ll mark eight onto our ๐‘ฆ-axis. And then at the point with coordinates one, eight, we add a hollow circle. Then if we connect these two points with a line segment, weโ€™ve sketched the line ๐‘ฆ is equal to eight ๐‘ฅ, where our values of ๐‘ฅ must be in the left-closed, right-open interval from zero to one. This means weโ€™ve successfully sketched our first subfunction.

Letโ€™s clear some space and then do the same to sketch our second subfunction. This time, our values of ๐‘ฅ will lie in the closed interval from one to seven. But this time we can see the output values of our function are a constant value of eight. This means when we sketch the graph of this subfunction, the ๐‘ฆ-coordinates of every point on our graph will be eight. Once again, we can find the endpoints of this subfunction. First, when ๐‘ฅ is equal to seven, we know ๐‘ฆ is going to be equal to eight. So our first endpoint has coordinates seven, eight. We draw this as a solid dot because our interval is closed on this side. And we have a very similar story when ๐‘ฅ is equal to one. Our ๐‘ฆ-coordinate will be equal to eight, and this interval is closed. So we draw a solid dot. We then connect these with a horizontal line to sketch our second subfunction.

And itโ€™s worth noting we have something interesting at the point one, eight. In our first subfunction we had a hollow dot at this point, but in our second function we had a solid dot at this point. Since there is a solid dot at this point, we know ๐‘“ evaluated at one is equal to eight. So this point is included in our graph. So we need to draw this as part of our graph. In other words, the solid dot takes over the hollow dot.

Letโ€™s now move on to our third subfunction. This time, our values of ๐‘ฅ are going to be in the left-open, right-closed interval from seven to 15. And once again, we have a linear function. So weโ€™ll do this by finding the endpoints of this subfunction. First, letโ€™s start by substituting ๐‘ฅ is equal to seven into our subfunction. We get the corresponding ๐‘ฆ-coordinate is 15 minus seven, which, we can calculate, is equal to eight. Therefore, the first endpoint of this subfunction has coordinates seven, eight. We should draw a hollow dot at this point on our graph. However, we can see the graph of our function already passes through this point, so we donโ€™t need to sketch this part onto our diagram. We just need to consider that this is the first endpoint of this subfunction.

Letโ€™s now find the second endpoint of this subfunction. We substitute ๐‘ฅ is equal to 15 into our subfunction to get the corresponding ๐‘ฆ-coordinate is 15 minus 15, which, we can calculate, is equal to zero. And remember, our subdomain is closed at the value of 15. 15 is in the domain of our function ๐‘“ of ๐‘ฅ. So we need to include this point on our graph. So we sketch this with a solid dot. Finally, we connect the two endpoints together with a line segment.

Now weโ€™ve sketched all three parts of our piecewise-defined function ๐‘“ of ๐‘ฅ. So this entire graph is just the function of ๐‘“ of ๐‘ฅ, where weโ€™ve included three different colors to highlight the three subfunctions. Now we can determine the range of this function from its graph. We just need to determine the set of all possible output values given its domain.

In the diagram, the output values of a function are the ๐‘ฆ-coordinates of any point on its curve. For example, on the graph, we can see the highest possible output of our function is eight. We can also see the lowest possible output. The lowest ๐‘ฆ-coordinate of any point on our curve is zero. Well, we notice when ๐‘ฅ is equal to zero and ๐‘ฅ is equal to 15, we have solid dots. So we know our curve passes through these points. And we could also see from the diagram any value of ๐‘ฆ between these two values is a possible output of our function. Therefore, the range of our function is all of the values between zero and eight. We can write this as the closed interval from zero to eight, which is our final answer.

Therefore, we were able to determine the range of a given piecewise linear function ๐‘“ of ๐‘ฅ by sketching its graph. We were able to show the range of this function was the closed interval from zero to eight.

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