Which of the following correctly describes the interval over which the illustrated function is negative? Is it (A) the open interval from negative ∞ to negative two? Is it (B) the open interval from two to ∞, (C) the open interval from negative ∞ to two, (D) the open interval from negative two to ∞? Or (E) is it the set of real numbers minus the set containing negative two?
And then we have a graph of the function plotted on the coordinate plane. In order to identify the correct set over which the function is negative, let’s simply remind ourselves what it means for the function to be negative. Let’s suppose the function is given by some expression 𝑓 of 𝑥. If the output of that function, the value that yields after we substitute any given value of 𝑥 in is negative, then we can say the function itself is negative. And of course, if we think about the output of a function, we think about the spread of values in the 𝑦-direction. Since the values of 𝑦 are negative if they lie underneath the 𝑥-axis, then we can say that the function itself 𝑓 of 𝑥 must be negative in this region.
So, let’s look carefully at the graph of our function. 𝑓 of 𝑥 is equal to zero when 𝑥 is equal to two. It’s less than zero for any values of 𝑥 greater than two. Note that we don’t actually include 𝑥 equals two in this definition; that’s because when 𝑥 is equal to two, the function is zero. This is neither negative nor positive at this point.
So, 𝑓 of 𝑥 is negative for all portions of the graph that lie underneath the 𝑥-axis or for all values of 𝑥 that are greater than two. Using set notation, we can say that that is the open interval from two to ∞. And in this case, that’s option (B).