Inverse Function

### Inverse Functions

A function can be viewed as a set of instructions. For example a mathematical function could double the input and then add one. We can write this function in function notation as:

(1)
\begin{equation} f(x)=2x+1 \end{equation}

If the function above was given an input of 3 the output would be 7. It seems likely that we could create a function to go backwards, that is if the input is 7 the output would be 3. Such a function, if it exists and they don't always, is called the inverse function. To create the inverse function of f we can go back to the written description and "undo it." We must do the inverse mathematical operations in the reverse order of the original function.

Function: double the input and then add one

Inverse Function: Subtract one from the input and then divide by 2

Note that subtraction and addition are inverses and division and and multiplication are inverses.

We can now write the inverse function in function notation:

(2)
\begin{align} f^{-1}(x)=\frac{x-1}{2} \end{align}

Try it… If the input to the inverse function is 7 the output is 3.

Note that the notation $f^{-1}$ is the name of the function and DOES NOT imply any mathematical operation. That is: $f^{-1}(x) \neq (f(x))^{-1}$. One notation implies the inverse the other implies the reciprocal!

### Finding Inverses Graphically

A sketch of an the inverse of a function can be drawn fairly quickly by reflecting the original graph across the line $y=x$ shown as the dotted line in the graphs below. If we assume that the red function is the original and the blue is the inverse function, we can notice the the x-intercepts of the original function "become" y-intercepts of the inverse and vice versa. This makes drawing inverse functions much easier as it gives a few easy to draw points on the graph.

Note that in the graph above the inverse of the function is itself still a function, that is the inverse passes the vertical line test. That is not always true as shown below. If a graph is given you can always draw the inverse, but the inverse is not always a function. If the inverse is itself a function then the original function is said to be "invertible." This is often confusing to students as it is possible to draw the inverse of a function but the function can still be non-invertible (as in the graph above).

### Finding Inverses Algebraically

Lets start with the example $y=\sqrt{2x+1}$. The first step is to swap the x and y this is essentially reflecting the function across the line $y=x$.

(3)
\begin{align} x=\sqrt{2y+1} \end{align}

Next we solve for y by step by step "unwrapping" the y. We must start on the outside of the function, in this case the square root, so we square both sides:

(4)
\begin{equation} x^2=2y+1 \end{equation}

Next we subtract 1 (reverse order of operations, nothing new):

(5)
\begin{equation} x^2-1=2y \end{equation}

And finally divide by 2:

(6)
\begin{align} y=\frac{x^2-1}{2} \end{align}

The result is the inverse function of the original function. This can be checked by graphing (looking to see it if is truly a reflection across $y=x$) or numerically by choosing an input for the original function such as 4:

(7)
\begin{align} y=\sqrt{2(4)+1)}=\sqrt{9}=3 \end{align}

Then using the output of the original as an input to the inverse:

(8)
\begin{align} y=\frac{(3)^2-1}{2}=\frac{8}{2}=4 \end{align}

We got the original number back so we did it correctly. While the example above is simple the process is the same for any function…