Dmitry B. answered • 04/07/20
Five years of Prealgebra tutoring at a math group
What does function actually do? In a very simple way you can say it converts an input into an output based on some rule. And this RULE is a function.
What does the inverse function do? The inverse function is just any other function, but it could behave as an inverse to some other function, which means it converts BACK the output to input. This is why it is INVERSE.
It is UNDO operation, CNTR-Z in term of computers. You publish your photo on Instagram. The "Publish" button is a function and "Delete" photo is an inverse function. Obviously this is just very general and simplified way to say. There are functions that don't have an INVERSE functions!!!
In any case, your function converts X = 1 to 3. So, what would be a function to convert it back 3 to 1?
Let's call it g. Thus, by definition:
g( f(x) ) = x
The record above says that we apply unknown function "g" to the known to us function "f" and we should get our "x" value back. You can restate this as: "What button on Instagram I should click to remove my picture from there and get one back to my comp?". This all it means.
Now is the most important part. What do we actually now to find this function?
We know:
1) Input. In our example it is 3. In case of Instagram example it is a picture that is published on Instagram.
2) Output. In our above example it is 1. In case of Instagram example it is your picture BACK to your comp.
3) Function. In our example it is 2 * X +1. In case of Instagram example it is your knowledge about how to mark a picture and which button to click to delete one.
In math this all comes to solving this equation:
x = 2 * f(x) +1,
where x is output, f(x) is input, and 2 * (input) + 1 is a function.
Now we solve it to find f(x):
x = 2 f(x) +1.
f (x) = ( x -1 ) / 2. <---- This is our INVERSE function "g".
Let's check. Feed 3, you got back (3 -1) / 2 = 1. See this 1 that we wanted to get back.
Done!
