Video Transcript
The graph of a function 𝑓
intersects the 𝑥-axis at negative 2.25 and the 𝑦-axis at 0.85. Which of the following coordinates
are necessarily those of a point on the graph of the inverse function of 𝑓?
Before we consider the given
coordinates, let’s first recall what 𝑓 inverse represents. What is the inverse function of
𝑓? Well, by definition, it is the
unique function 𝑓 inverse such that 𝑓 inverse of 𝑓 of 𝑥 is equal to 𝑥 for all
𝑥.
Consider now the graph of 𝑓. We don’t actually know what it
looks like, but in any case it is the collection of all points of the form 𝑥, 𝑓 of
𝑥. The graph of 𝑓 inverse on the
other hand, consists of all points of the form 𝑥, 𝑓 inverse of 𝑥. Suppose we look now at a point on
the graph of 𝑓 inverse whose 𝑥-coordinate is of the form 𝑓 of 𝑥 for some 𝑥. What will the 𝑦-coordinate of this
point be? Well, it must have the form 𝑓
inverse of 𝑓 of 𝑥. However, by the definition of the
inverse function, this is nothing other than 𝑥.
We have seen that if we are given a
point 𝑥, 𝑓 of 𝑥 on the graph of 𝑓, we know that the point 𝑓 of 𝑥, 𝑥 must
exist on the graph of 𝑓 inverse. In general, points 𝑎, 𝑏 on the
graph of 𝑓 correspond to points 𝑏, 𝑎 on the graph of 𝑓 inverse. That is, coordinates of points on
the 𝑓 inverse graph are just coordinates of points on the 𝑓 graph with the 𝑥- and
𝑦-coordinates swapped. Graphically, this means that the
graph of 𝑓 inverse is simply the graph of 𝑓 reflected in the line 𝑦 equals
𝑥.
We were given two points on the
graph of 𝑓: the 𝑥-intercept at negative 2.25, zero and the 𝑦-intercept at zero,
0.85. By swapping the 𝑥- and
𝑦-coordinates of these points, we find two points that must lie on the graph of 𝑓
inverse: an 𝑥-intercept at zero, negative 2.25 and a 𝑦-intercept at 0.85,
zero.
Returning to the options for points
on the graph of 𝑓-inverse we were given, we can see that the only one guaranteed to
be present is 0.85, zero.
