Video Transcript
Which of the following functions represents the graph shown? Is it (A) 𝑓 of 𝑥 equals 𝑥, (B) 𝑓 of 𝑥 equals negative 𝑥, (C) 𝑓 of 𝑥 equals negative one, (D) 𝑓 of 𝑥 equals one, or (E) 𝑓 of 𝑥 equals zero?
There are a number of ways to identify which of the functions represents the graph shown. One technique is to identify the coordinates of each point on the graph. Given a rule in function notation that is 𝑓 of 𝑥 equals some function of 𝑥 or a constant, the coordinates of each point on the graph are given by input–output, that is, 𝑥, 𝑓 of 𝑥. So let’s identify a table of values for the function 𝑓 of 𝑥 equals 𝑥.
We know that the input 𝑥 corresponds to an output 𝑓 of 𝑥. So let’s choose three distinct values of 𝑥. We’ll choose negative two, zero, and two. But we could have chosen any values, for example, zero, one, and two. The function says that if we substitute 𝑥 in, we get 𝑥 out. So if we substitute negative two in, we get negative two out. Similarly, if we substitute zero, we get zero and if we substitute two, 𝑓 of two is two. So the three pairs of coordinates we can look for are negative two, negative two; zero, zero; and two, two. We plot those three coordinates on the graph, and we see that the red line passes straight through them. And so, the answer must be (A). 𝑓 of 𝑥 equals 𝑥 represents the graph shown.
Let’s double-check by adding (B), (C), (D), and (E) to the diagram. The second function 𝑓 of 𝑥 equals negative 𝑥 says that if we substitute 𝑥 in, we get negative 𝑥 out. So in the case of our table, if we substitute negative two into the function, we get negative negative two out, which is just two. If we substitute zero in, we get negative zero, which is just zero. And if we substitute two in, we get negative two out. We can add this to our diagram, and we can see in fact this is the graph of the original function reflected in the 𝑥-axis.
Let’s now think about the function 𝑓 of 𝑥 equals negative one. This tells us that for any value of 𝑥 we substitute in, we’ll always get negative one out. And so, we must have a horizontal line that has 𝑦-coordinates of negative one throughout. In a similar way, the graph of 𝑓 of 𝑥 equals one will also be a horizontal line that passes through one on the 𝑦-axis. And 𝑓 of 𝑥 equals zero will be a horizontal line passing through zero. In other words, it’s the 𝑥-axis. And so we have demonstrated that the function that represents the graph we were given is (A): 𝑓 of 𝑥 equals 𝑥.
