Question Video: Determining the Correct Representation of a Piecewise-Defined Function from Its Graph and Evaluating the Function at a Given Point Mathematics

Video Transcript

The function 𝑓 of 𝑥 can be represented by the graph shown below. Which of the following is the piecewise representation of 𝑓 of 𝑥? Is it option (A) 𝑓 of 𝑥 is equal to 𝑥 squared minus one when 𝑥 is greater than or equal to negative two and 𝑥 is less than four and 𝑓 of 𝑥 is equal to 25 when 𝑥 is greater than four and 𝑥 is less than or equal to 10 and 𝑓 of 𝑥 is equal to two 𝑥 plus eight when 𝑥 is greater than 10? Or is it option (B) 𝑓 of 𝑥 is equal to 𝑥 squared minus one when 𝑥 is greater than or equal to negative two and 𝑥 is less than four and 𝑓 of 𝑥 is equal to 25 when 𝑥 is greater than or equal to four and 𝑥 is less than or equal to 10 and 𝑓 of 𝑥 is equal to two 𝑥 plus eight when 𝑥 is greater than 10? Or is it option (C) 𝑓 of 𝑥 is equal to 𝑥 squared minus one when 𝑥 is greater than or equal to negative two and 𝑥 is less than four and 𝑓 of 𝑥 is equal to 25𝑥 when 𝑥 is greater than four and 𝑥 is less than or equal to 10 and 𝑓 of 𝑥 is equal to two 𝑥 plus eight when 𝑥 is greater than 10? Is it option (D) 𝑓 of 𝑥 is equal to 𝑥 squared plus one when 𝑥 is greater than or equal to negative two and 𝑥 is less than four and 𝑓 of 𝑥 is equal to 25 when 𝑥 is greater than four and 𝑥 is less than or equal to 10 and 𝑓 of 𝑥 is equal to two 𝑥 plus eight when 𝑥 is greater than 10. Or is it option (E) 𝑓 of 𝑥 is equal to 𝑥 squared plus one when 𝑥 is greater than or equal to negative two and 𝑥 is less than four, 𝑓 of 𝑥 is equal to 25 when 𝑥 is greater than four and less than 10, and 𝑓 of 𝑥 is equal to two 𝑥 plus eight when 𝑥 is greater than or equal to 10. Find the value of 𝑓 evaluated at three.

There are two parts of this question. In the first part of this question, we need to determine which of the five given piecewise representations of 𝑓 of 𝑥 is the correct piecewise representation of 𝑓 of 𝑥. In the second part of this question, we need to determine the value of 𝑓 at three. And there are many different ways of answering the first part of this question. The easiest way is to start by considering how would we construct a piecewise representation of 𝑓 of 𝑥 from its graph.

And there are three sections to this graph, so we would expect three subfunctions for our piecewise-defined function. And the easiest of these subfunctions to find is the second subfunction because it’s a constant value of 25. We want to find the subdomain for the second subfunction. That’s the values of 𝑥 we input into our piecewise-defined function to output the second part of our function.

On our graph, this will be the 𝑥-coordinate of any point on the second part of our function. We can see these lie between four and 10. And we need to know on the diagram we can see on the left side when 𝑥 is equal to four, we have a hollow circle, and on the right side when 𝑥 is equal to 10, we have a solid circle. When it’s hollow, we don’t include the endpoint, and when it’s solid, we do include the endpoint. So, our subdomain will be the values of 𝑥 greater than four and the values of 𝑥 less than or equal to 10.

This gives us our first subfunction. It’s a constant value of 25 and the values of 𝑥 must be greater than four and less than or equal to 10. Although it’s not necessary, we can use this to eliminate some of our options. First, we can see in option (B) our second subdomain includes the value of four. However, in the diagram, when 𝑥 is equal to four, our second subfunction has a hollow dot. So, four is not in the subdomain of our second subfunction. So, option (B) is not correct. In option (C), we can see the second subfunction is 25𝑥. However, we know it’s a constant value of 25. So, (C) is not the correct option either. Finally, in option (E), we can see that 10 is not included in the second subdomain. However, in the diagram, when 𝑥 is equal to 10, we can see we have a solid dot in the second subfunction. So, 10 is in the second subdomain of our function. So, our answer can’t be option (E).

We can do the same to find expressions for the first and third subfunctions and subdomains. Since the third subfunction is a linear function, this one will be easier. So, let’s do this one next. We’ll start by finding the subdomain of this subfunction. We can see that the graph of this subfunction starts when 𝑥 is equal to 10, and it continues indefinitely. And since the graph of this subfunction has a hollow dot at its endpoint, we don’t include 𝑥 is equal to 10 in the subdomain. The subdomain of this function is just all values of 𝑥 greater than 10.

Next, we need to find an expression for this linear function. There’s a few different ways of doing this. One way is to find the slope and the coordinates of a point on the line. We can find the slope of this line directly from the graph. For every one unit we move across, we move two units upwards. The slope of this line is two. We can then find an equation from this line, either by extending the line and finding its 𝑦-intercept or by using the coordinates of a point on the line. We see that our line passes through the point 13, 34. This allows us to find the equation of this line by using the point–slope form of the equation of a line.

A line with slope 𝑚 passing through the point with coordinates 𝑥 one, 𝑦 one will have the equation 𝑦 minus 𝑦 one is equal to two times 𝑥 minus 𝑥 one. In this case, that’s 𝑦 minus 34 is equal to two multiplied by 𝑥 minus 13. And if we distribute, rearrange, and simplify this equation, we get the equation of this line is 𝑦 is equal to two 𝑥 plus eight. Therefore, the third subfunction of our piecewise-defined function is two 𝑥 plus eight with the subdomain 𝑥 being greater than 10.

However, we can just notice both of our remaining options already have this as the third subfunction and third subdomain. But it is useful to be able to determine subfunctions and subdomains from a graph because we won’t always be given the options. Instead, let’s clear some space and determine the first subfunction and subdomain. Once again, we’ll start by determining the subdomain of this subfunction. We see the values of 𝑥 for this part of the graph range from negative two to four. And there’s a solid circle when 𝑥 is equal to negative two and a hollow circle when 𝑥 is equal to four. This means the subdomain will include the value of negative two, and it won’t include the value of four. So, our subdomain is 𝑥 is greater than or equal to negative two and 𝑥 is less than four.

All that’s left to do is to determine the first subfunction. There’s a few different ways of doing this. The easiest way is to notice that this has the same shape as a quadratic parabola. So, if we sketch 𝑦 is equal to 𝑥 squared on the closed interval from negative two to four onto our diagram, we can see that our subfunction is translated one unit down from this parabola. And to translate a function down, we subtract one from its output. So, our first subfunction is 𝑥 squared minus one.

This gives us the full piecewise representation of our function 𝑓 of 𝑥. And we can see this is given by option (A) 𝑓 of 𝑥 is equal to 𝑥 squared minus one when 𝑥 is greater than or equal to negative two and less than four, 𝑓 of 𝑥 is equal to 25 when 𝑥 is greater than four and less than or equal to 10, and 𝑓 of 𝑥 is equal to two 𝑥 plus eight when 𝑥 is greater than 10.

But we’re not done yet. The second part of our question wants us to determine 𝑓 evaluated at three. And there’s two different ways we can do this. Let’s start by clearing some space and do this from the diagram. In a graph, the 𝑥-coordinates of any point on our curve tell us the input values of the function and the corresponding 𝑦-coordinates tell us the output of our function. So, we can determine 𝑓 evaluated at three by reading off the 𝑦-coordinate of the point on the curve with 𝑥-coordinate three. And from the graph, we can see this is eight. Therefore, 𝑓 evaluated at three is equal to eight.

However, this isn’t the only way we can answer this question. We can also do this directly from the piecewise definition of 𝑓 of 𝑥 we determined in the first part. To input the value of three into this function, we need to determine which subdomain the value of three lies in. And three is bigger than negative two and less than four, so three lies in the first subdomain of our function. Therefore, we evaluate 𝑓 at three by substituting 𝑥 is equal to three into our first subfunction. 𝑓 of three is three squared minus one, which is also equal to eight. Therefore, 𝑓 evaluated at three is equal to eight.