At the most basic definition, inverse functions are simply the opposite of another function. Instead of f(x) for the function, an inverse function is f^-1(y). Inverse functions are not available for every function, but when they are available, it is possible to reverse the original function to determine what the inverse function is.
Reversing a Function
Creating an inverse is done by simply reversing the original function. This can be done both ways, so even after an inverse is created, it’s possible to go back to the original function as well. If the original function is f(x)=2x-7, the order for the function is to multiply the x by 2 and then subtract 7. The inverse reverses this, so it adds 7 to the y and then divides by 2. So, the inverse of f(x)=2X-7 is f^-1(y)=(y+7)/2.
Checking to See if the Inverse is Correct
It’s important to make sure the inverse that is created is correct. This is done by filling in a number for x and solving the original function. Then, use the answer for that as y in the inverse to see if the original number is the answer.
For instance, use the function f(x)=3x+5. First, determine the inverse of the function. This would be f^-1(y)=(y-5)/3 because the inverse function is the reverse of the function. Then, solve for x=3. The original function would then be f(3)=14. To see if the inverse is correct, solve it using y=14. So, f^-1(14)=(14-5)/3. This does solve as f^-1(14)=3. Since the number from the inverse function matches the number used to solve the function, the inverse was done correctly and the inverse is valid.
Another Way to Find the Inverse
Sometimes, it’s not easy to simply reverse the function and find the inverse. In these cases, it’s important to use algebra to easily find the inverse. Substitute y for f(x) and then solve for x. Then, replace the x with f^-1(y) for the inverse.
F(x)=2x+5
Y=2x+5
y-5=2x
(y-5)/2=x
F^-1(x)=(y-5)/2
Once the inverse is found, it’s possible to use the steps above to ensure the inverse is correct and valid. There are times when this does not work properly and an inverse cannot be found. Some expressions simply do not have an inverse.
Inverse functions are simply reversed functions and it can be easy to determine what the inverse is of a function as long as it has an inverse. It’s important to be careful when using either method to find the inverse as the opposite symbol will need to be used for the inverse. If the original function had addition, it would become subtraction in the inverse. If there was originally division in the function, it would become multiplication in the inverse and so on. As long as the inverse is valid, using the above method for solving the function and the inverse will enable the student to check to ensure the work was done properly.
Functions and inverse functions are useful outside of math class as well. Perhaps a student knows how to convert Celsius to Fahrenheit and they’re given a temperature in Fahrenheit they need know in Celsius. Knowing how to find the inverse function will allow them to reverse the formula they already know to find the one they need.