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Find the range of the function 𝑓 of 𝑥 is equal to eight 𝑥 if 𝑥 is in the left-closed, right-open interval from zero to one and 𝑓 of 𝑥 is equal to eight if 𝑥 is in the closed interval from one to seven and 𝑓 of 𝑥 is equal to 15 minus 𝑥 if 𝑥 is in the left-open, right-closed interval from seven to 15.
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In this question, we’re asked to find the range of a given piecewise function.
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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.
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And there’s many different ways of finding the range of a function.
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Since we’re given a piecewise-defined function where each of the three subfunctions are linear, we’ll do this by sketching its graph.
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Before we sketch our graph, let’s determine the domain of this function.
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That’s the set of possible input values for our function.
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To do this, we know we have a piecewise-defined function.
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And the domain of any piecewise-defined function is the union of its subdomains.
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In other words, we can only input values of 𝑥 into our function which are in these three sets.
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So we’ll start by sketching our coordinate axes, where on the 𝑥-axis, we need to mark the endpoints of our subdomains.
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That’s zero, one, seven, and 15.
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And it is worth noting we may need to extend this to include negative values of 𝑦.
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However, we’ll see in this case it’s not necessary.
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We now need to sketch each subfunction separately over its subdomain.
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Let’s start with the first subfunction defined over the left-closed, right-open interval from zero to one.
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We can see that this function is the linear function eight 𝑥.
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Since this is a linear function defined over an interval, this will be a line segment.
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And the easiest way to sketch a line segment is to find the coordinates of its two endpoints.
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To find the endpoints of this line segment, we need to substitute the endpoints of our subdomain into the subfunction.
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Let’s start by substituting 𝑥 is equal to zero into our subfunction.
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We get eight multiplied by zero, which is equal to zero.
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Since zero is in the subdomain of this function, this tells us that 𝑓 evaluated at zero is equal to zero, which in turn tells us the graph of our function passes through the origin.
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We’ll mark this with a solid dot.
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We now want to check the other endpoint of our subdomain.
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However, we do need to notice that this side of our interval is open.
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This means we can’t evaluate 𝑓 at one by substituting it into the subfunction eight 𝑥.
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However, we can use this to find the other endpoint of our subfunction.
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Substituting 𝑥 is equal to one into the subfunction eight 𝑥, we get eight multiplied by one, which is equal to eight.
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This is then the 𝑦-coordinate of the endpoint of our first subfunction.
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The endpoint of our subfunction will be one, eight.
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So we’ll mark eight onto our 𝑦-axis.
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And then at the point with coordinates one, eight, we add a hollow circle.
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Then if we connect these two points with a line segment, we’ve sketched the line 𝑦 is equal to eight 𝑥, where our values of 𝑥 must be in the left-closed, right-open interval from zero to one.
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This means we’ve successfully sketched our first subfunction.
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Let’s clear some space and then do the same to sketch our second subfunction.
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This time, our values of 𝑥 will lie in the closed interval from one to seven.
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But this time we can see the output values of our function are a constant value of eight.
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This means when we sketch the graph of this subfunction, the 𝑦-coordinates of every point on our graph will be eight.
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Once again, we can find the endpoints of this subfunction.
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First, when 𝑥 is equal to seven, we know 𝑦 is going to be equal to eight.
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So our first endpoint has coordinates seven, eight.
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We draw this as a solid dot because our interval is closed on this side.
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And we have a very similar story when 𝑥 is equal to one.
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Our 𝑦-coordinate will be equal to eight, and this interval is closed.
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So we draw a solid dot.
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We then connect these with a horizontal line to sketch our second subfunction.
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And it’s worth noting we have something interesting at the point one, eight.
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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.
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Since there is a solid dot at this point, we know 𝑓 evaluated at one is equal to eight.
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So this point is included in our graph.
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So we need to draw this as part of our graph.
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In other words, the solid dot takes over the hollow dot.
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Let’s now move on to our third subfunction.
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This time, our values of 𝑥 are going to be in the left-open, right-closed interval from seven to 15.
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And once again, we have a linear function.
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So we’ll do this by finding the endpoints of this subfunction.
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First, let’s start by substituting 𝑥 is equal to seven into our subfunction.
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We get the corresponding 𝑦-coordinate is 15 minus seven, which, we can calculate, is equal to eight.
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Therefore, the first endpoint of this subfunction has coordinates seven, eight.
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We should draw a hollow dot at this point on our graph.
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However, we can see the graph of our function already passes through this point, so we don’t need to sketch this part onto our diagram.
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We just need to consider that this is the first endpoint of this subfunction.
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Let’s now find the second endpoint of this subfunction.
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We substitute 𝑥 is equal to 15 into our subfunction to get the corresponding 𝑦-coordinate is 15 minus 15, which, we can calculate, is equal to zero.
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And remember, our subdomain is closed at the value of 15.
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15 is in the domain of our function 𝑓 of 𝑥.
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So we need to include this point on our graph.
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So we sketch this with a solid dot.
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Finally, we connect the two endpoints together with a line segment.
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Now we’ve sketched all three parts of our piecewise-defined function 𝑓 of 𝑥.
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So this entire graph is just the function of 𝑓 of 𝑥, where we’ve included three different colors to highlight the three subfunctions.
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Now we can determine the range of this function from its graph.
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We just need to determine the set of all possible output values given its domain.
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In the diagram, the output values of a function are the 𝑦-coordinates of any point on its curve.
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For example, on the graph, we can see the highest possible output of our function is eight.
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We can also see the lowest possible output.
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The lowest 𝑦-coordinate of any point on our curve is zero.
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Well, we notice when 𝑥 is equal to zero and 𝑥 is equal to 15, we have solid dots.
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So we know our curve passes through these points.
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And we could also see from the diagram any value of 𝑦 between these two values is a possible output of our function.
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Therefore, the range of our function is all of the values between zero and eight.
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We can write this as the closed interval from zero to eight, which is our final answer.
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Therefore, we were able to determine the range of a given piecewise linear function 𝑓 of 𝑥 by sketching its graph.
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We were able to show the range of this function was the closed interval from zero to eight.