Video Transcript
The diagram shows the graph of 𝑦
is equal to 𝑓 of 𝑥. What is the solution set of the
equation 𝑓 of 𝑥 is equal to zero?
In this question, we’re given the
graph of a function 𝑦 is equal to 𝑓 of 𝑥. And we’re asked to use this graph
to determine the solution set of the equation 𝑓 of 𝑥 is equal to zero. Before we start answering this
question, we can note that the graph 𝑦 is equal to 𝑓 of 𝑥 has a parabolic shape
which opens upwards. This means it’s likely that 𝑓 of
𝑥 is a quadratic function with positive leading coefficient. However, this is not strictly
necessary. And we don’t need this information
to answer the question. Instead, we can start by recalling
the solution set of an equation is the set of all solutions to the equation. In this case, it will be the set of
all values of 𝑥 which satisfy the equation 𝑓 evaluated at 𝑥 is equal to zero.
In other words, we’re looking for
all of the values of 𝑥 where the output of the function is zero. And we can find these 𝑥-values by
using the given diagram. We recall when we graph a function,
the 𝑥-coordinate of any point on the curve tells us the input value of the function
and the corresponding 𝑦-coordinate tells us the output value of the function. For example, we can see the point
with coordinates two, five lies on the graph of 𝑦 is equal to 𝑓 of 𝑥. The 𝑥-coordinate of this point is
the input value of the function, and the 𝑦-coordinate is the corresponding
output. 𝑓 evaluated at two must be equal
to five.
We want to find the input values of
the function where the output of the function is zero. So we want to find the points on
the curve whose 𝑦-coordinate is equal to zero. And the 𝑦-coordinate will be equal
to zero whenever the function crosses the 𝑥-axis. In other words, the solutions to
this equation are the 𝑥-intercepts of the curve.
And we can see that there are two
𝑥-intercepts of this parabola: negative three and one. In particular, 𝑓 evaluated at
negative three is equal to zero and 𝑓 evaluated at one is also equal to zero
because the 𝑦-coordinate of these points on the curve is zero. And remember, we need to write this
as the set of all 𝑥-values which solve the equation. So, that’s the set containing
negative three and one. And it’s worth noting that this
method holds in general. The solution set of the equation 𝑓
of 𝑥 is equal to zero is always the same as the set of 𝑥-intercepts of the graph
of the function 𝑦 is equal to 𝑓 of 𝑥.
Therefore, we were able to show by
using the diagram of the graph of 𝑦 is equal to 𝑓 of 𝑥, the solution set of the
equation 𝑓 of 𝑥 is equal to zero is the set containing negative three and one.
