Question Video: Finding the Piecewise Definition of a Function from Its Graph Mathematics • Second Year of Secondary School

Video Transcript

Which of the following is the piecewise definition of the function 𝑓 of 𝑥 whose graph is shown? Is it 𝑓 of 𝑥 is equal to 𝑥 plus three when 𝑥 is greater than negative eight and 𝑥 is less than or equal to negative three? And 𝑓 of 𝑥 is equal to 𝑥 squared minus nine when 𝑥 is greater than negative three and 𝑥 is less than two. And 𝑓 of 𝑥 is equal to negative five when 𝑥 is greater than or equal to two and 𝑥 is less than six. Is it option (B) 𝑓 of 𝑥 is equal to 𝑥 plus three when 𝑥 is greater than negative eight and 𝑥 is less than or equal to negative three? And 𝑓 of 𝑥 is equal to 𝑥 squared minus three when 𝑥 is greater than negative three and 𝑥 is less than two. And 𝑓 of 𝑥 is equal to negative five when 𝑥 is greater than or equal to two and 𝑥 is less than six. Or is it option (C) 𝑓 of 𝑥 is equal to 𝑥 plus three when 𝑥 is greater than negative eight and 𝑥 is less than or equal to negative three? And 𝑓 of 𝑥 is equal to 𝑥 squared minus nine when 𝑥 is greater than negative three and less than two. And 𝑓 of 𝑥 is equal to negative five 𝑥 when 𝑥 is greater than or equal to two and less than six. Option (D) 𝑓 of 𝑥 is equal to 𝑥 plus two when 𝑥 is greater than negative eight and less than or equal to negative three. And 𝑓 of 𝑥 is equal to 𝑥 squared minus nine when 𝑥 is greater than negative three and less than two. And 𝑓 of 𝑥 is equal to negative five when 𝑥 is greater than or equal to two and less than six. Or finally is it option (E) 𝑓 of 𝑥 is equal to 𝑥 plus three when 𝑥 is greater than negative eight and less than or equal to negative three? 𝑓 of 𝑥 is equal to 𝑥 squared minus three when 𝑥 is greater than negative three and less than two. And 𝑓 of 𝑥 is equal to negative five 𝑥 when 𝑥 is greater than two and less than six.

In this question, we’re given a graph of a function and we need to determine which of five given options is the correct piecewise definition of the graphed function. And there’s many different ways of going about answering this question. For example, we could sketch the five given options and see which one has the same graph as the given function. Alternatively, we could eliminate options by considering the graphs of these functions and considering the given graph. And both of these methods would work and would give us the correct answer. However, they do rely on us knowing the given options.

So instead, let’s try and answer this question without using the given options. We can do this by using the graph of the function to determine a piecewise definition of this function. So first, we need to look at the graph and see that it splits into three subfunctions: the first subfunction in green, which is a straight line; the second subfunction in blue, which is a parabolic shape; and the third subfunction in red, which is a horizontal line. Let’s find each of these subfunctions and the subdomains. Let’s start with the first subfunction.

To determine the subdomain of this subfunction, we need to find the input values of 𝑥 from the graph. And we can see in the graph, these input values range from negative eight to negative three. However, we do need to be careful since when 𝑥 is equal to negative three, the graph has a hollow dot. This means negative eight is not included in the subdomain. However, we have a choice whether we want to include negative three in this subdomain or the second subdomain. Either would work. However, in the five given options, all of them have negative three included in the first subdomain. So we’ll include negative three in the first subdomain, giving us 𝑥 must be bigger than negative eight and less than or equal to negative three.

We now need to find the equation of this line to find the first subfunction, and there’s a few different ways of doing this. We’re going to use the slope–intercept form of a line. First, let’s find the slope from the diagram. We can see as we move one unit to the right, the line moves one unit up, so the slope of this function is one. Next, we can extend the line to find the 𝑦-intercept. The 𝑦-intercept is at three, so our value of 𝑏 in the slope–intercept form will be three. And it’s also worth pointing out we could’ve found this by noting as we move three units to the right, since the slope is one, we need to move three units up. Therefore, this is a straight line with slope one and 𝑦-intercept three. In the slope–intercept form, this function will be 𝑥 plus three.

Let’s now move on to the second subfunction. We need to determine the equation of this parabola. We’ll do this by using the fact we’re given the coordinates of the vertex of this parabola. And we can see in the diagram, this is the point zero, negative nine. We can then use the coordinates of the vertex and the vertex form of a parabola to find an equation for this parabola. Since the 𝑥-coordinate of the vertex is zero and the 𝑦-coordinate is negative nine, we get 𝑦 is equal to 𝑘 multiplied by 𝑥 minus zero squared minus nine. And it’s also worth pointing out since we can see in the diagram the parabola opens upwards, we can conclude that 𝑘 must be positive. We can then simplify the right-hand side of this equation. We get 𝑦 is equal to 𝑘𝑥 squared minus nine.

To determine the second subfunction, we’re going to need to find the value of 𝑘. We can do this by substituting in the coordinates of any other point which lies on the parabola. We’ll choose the given 𝑥-intercept, the point negative three, zero. Substituting these coordinates in, we get zero is equal to 𝑘 times negative three squared minus nine. And we can then solve this equation for 𝑘. Simplifying, we get zero is equal to nine 𝑘 minus nine. We can then add nine to both sides of the equation and divide through by nine to see that 𝑘 must be equal to one. So we can substitute 𝑘 is equal to one, and this means our second subfunction must be 𝑥 squared minus nine.

We can then find the second subdomain in exactly the same way. We need to use the diagram to determine the input values of 𝑥 for this section of the graph. And remember, we already chose negative three to be included in the first subdomain, so it won’t be included in the second subdomain. We can then see in the diagram the 𝑥-values for the parabola range from negative three to two. We’ve already noted we’re not including negative three in this subdomain. However, we can see in the five given options, two is not included in any of the second subdomains. So we’ll also choose not to include two in this subdomain. We get 𝑥 must be greater than negative three and 𝑥 is less than two. Of course, this means we are going to have to include two in the third subdomain.

So let’s start by finding the third subdomain of the third subfunction. Once again, we need to find the 𝑥-values for this section of the graph by using a diagram. We decided we’re going to include the value of two. However, we can see in the diagram there’s a hollow circle in the diagram when 𝑥 is equal to six, so we’re not including the value of six. However, we are including all the values greater than or equal to two and less than six. Therefore, our third subdomain is the values of 𝑥 greater than or equal to two and less than six.

All that’s left to do is find the third subfunction, and we can do this directly from the diagram. It’s a horizontal line at negative five. So the outputs are a constant value of negative five, so the third subfunction is just negative five. Therefore, we were able to show the piecewise definition of the graphed function is 𝑓 of 𝑥 is equal to 𝑥 plus three when 𝑥 is greater than negative eight and less than or equal to negative three. And 𝑓 of 𝑥 is equal to 𝑥 squared minus nine when 𝑥 is greater than negative three and less than two. And 𝑓 of 𝑥 is equal to negative five when 𝑥 is greater than or equal to two and less than six. And this was given as option (A).