### Video Transcript

Which of the following graphs represents the function π of π₯ equals π₯ squared when π₯ is less than two and negative two π₯ plus 10 when π₯ is greater than or equal to two?

The function in the question is defined differently over different sections of its domain. When π₯ is less than two, π of π₯ is equal to the quadratic function π₯ squared, whereas when π₯ is greater than or equal to two, π of π₯ is equal to the linear function negative two π₯ plus 10. π of π₯ is therefore a piecewise-defined function, that is, a function that consists of multiple subfunctions, each of which applies to a given subdomain. We need to identify the graph of this piecewise function. So letβs have a go at sketching it ourselves.

As weβve already said, for values of π₯ strictly less than two, π of π₯ is equal to the quadratic function π₯ squared. We can sketch this either from our knowledge of quadratic graphs or by substituting some π₯-values less than two. For example, when π₯ is equal to negative two, π₯ squared is equal to four. When π₯ is equal to positive or negative one, π₯ squared is equal to one. And when π₯ is equal to zero, π₯ squared is also equal to zero. So we obtain this part of the graph. Now, because this part of the definition is only true for π₯-values strictly less than two and not when π₯ is equal to two itself, letβs put an open circle at the right-hand end of this quadratic curve to indicate that weβre not including the point two, four.

Weβve drawn part of the graph of the function π of π₯ then. Letβs think now about what happens when π₯ is greater than or equal to two. π of π₯ is equal to negative two π₯ plus 10, which is a linear function, so the graph is a straight line. The easiest way to sketch this is perhaps to substitute some π₯-values. When π₯ is equal to two, π of π₯ is equal to negative two multiplied by two plus 10. Thatβs negative four plus 10, which is equal to six, so we have the point with coordinates two, six. When π₯ is equal to four, π of π₯ is equal to negative two multiplied by four plus 10. Thatβs negative eight plus 10, which is equal to two. So we also have the point four, two.

Plotting these points and then joining them with a straight line gives the second portion of the graph. And this time, because this part of the definition of the function does include π₯ equals two, we have a solid dot at the point where π₯ equals two.

Letβs now compare what weβve drawn with the five graphs we were given. And when we do, we can see that theyβre all very similar. All five graphs have a quadratic section when π₯ is less than two and then a linear section with a negative slope. If we look at graphs (A), (B), and (E), however, we can see that the linear sections sketched here do not correspond to the function π of π₯ equals negative two π₯ plus 10. We know that this function includes the point with coordinates two, six. And in fact, there should be a vertical gap between the two sections of the graph. So we can rule out options (A), (B), and (E). And weβre left with graphs (C) and (D). These are very, very similar. The only difference is in what happens when π₯ is equal to two. In graph (C), there is a solid dot on the quadratic section and an open dot on the linear section, whereas on graph (D), the reverse is true.

Looking back at our graph, we can see that the solid dot should be on the linear section because π₯ equals two is included in the second part of the functionβs subdomain. So graph (D) is the correct graph for this piecewise function. It has the correct quadratic function when π₯ is less than two, the correct linear function when π₯ is greater than or equal to two, and the correct combination of closed and open circles at the value of π₯ where the functionβs definition changes.