There are a couple of ways to think about the inverse
of a function. We can approach inverses by looking at graphs or performing algebraic operations. In either case, it comes down to the basic notion that the inverse
of a function
reverses the x and y coordinates. In other words, for every ordered pair
in a function
there will be an ordered pair
in the inverse
When we look at a graph, a function
is reflected over the line
to create the inverse
of the function. By reflecting over the line
we are achieving the goal of reversing the x and y coordinates.
In the graph
below, the original function
is reflected over the line
(which is shown as a dotted line) and gives us the inverse function
notation indicates that we are talking about the inverse
of a function.
A graphical approach is helpful to:
- show that two functions are inverses of each other or not
- sketch the inverse of a function by reflecting it over the line
above displayed two functions that were inverses of each other. We were told what those two functions were and could look at the graph
and see that they are inverses of each other. But where did those two functions come from? If we are given just an original function, how do we go about finding an inverse
on our own? It goes back to the idea of reversing the x and y coordinates.
Let’s return and re-examine the function
. What is the process used to find the inverse? A step by step process is shown below.
- Write the original function using y = notation. Remember that is just the name of our function and is often used interchangeably with y. So we write as
- Interchange the x and y. Remember this is the foundation behind an inverse. So the equation will now become
- Solve the new equation for y. Remember that equations are usually easier to deal with if we have y on one side and everything else on the other side. In solving for y, we get
We then simplify this equation to
- Change the y to inverse notation. This step just helps to ensure that we clearly indicate the inverse. Back in Steps 2 and 3, we had another y = equation and this step just makes sure we don’t have too much confusing notation. So we end up with the inverse as . This is a good thing since we already showed in the graph that the two functions are inverses.
Let’s use this process when we don’t already know the answer and find the inverse
- Change to y.
- Interchange x and y.
- Solve for y.
- Change to inverse notation.
We now have a four step process to find the inverse
of a given function. In the first example we did, we already knew the answer to confirm our process was correct. But in the second example we don’t already know what the answer is supposed to be. How do you know if two functions are indeed inverses of each other? One way is by looking at the graphs. Another way is algebraically using composition of functions. If you are not familiar with composition of function, click here
to learn more.
We have a function
and a candidate for the inverse
. We can show these two functions really are inverses of each other by finding
AND making sure those two compositions are equal to x
Let’s begin by finding
This is half of our process. We also have to verify
is also equal to x.
We have now confirmed that
are inverses of each other.
If we graphed these two functions we could see that they are reflections of each other over the line
. (Think about folding the graph
along the line
and noticing that the function
and its inverse
are superimposed over each other.)