Video Transcript
The graph shows the function 𝑓 of
𝑥 equals 𝑥 squared minus two 𝑥 plus three. What is the solution set of 𝑓 of
𝑥 equals zero.
Given some graph of a function 𝑦
equals 𝑓 of 𝑥, the solutions to the equation 𝑓 of 𝑥 equals zero correspond to
the values of the 𝑥-intercepts if they exist. If there are no 𝑥-intercepts, it
follows that there are no solutions to the equation 𝑓 of 𝑥 equals zero. Now in this case, we’ve been given
the graph of the function 𝑓 of 𝑥 equals 𝑥 squared minus two 𝑥 plus three and
asked to solve 𝑓 of 𝑥 equals zero or, in other words, 𝑥 squared minus two 𝑥 plus
three equals zero.
But if we look carefully, we see
that there are no locations where the graph, which is the green plot here,
intersects the 𝑥-axis. And so, there are no real solutions
to the equation 𝑓 of 𝑥 equals zero. In order to use set notation, we
need to find a way to show that there are no values inside the set. So we use the notation shown. It’s the null set or empty set.
