2.10 F) Inverse Functions
A function is something that takes an input value and produces an output. The function f(x) is given below.
An inverse function is a function that links an output value to an input value; essential an inverse function works in the opposite direction to a function. We denoted inverse functions with a superscript -1. For example, the inverse function for f(x) would be denoted as f-1(x).
The easiest way to find the inverse function is to let y equal the function. For the f(x) example we would get the equation that is given below.
The next step is to swap the x and y values*.
We then make y the subject of the formula. The first step in making y the subject of the formula is to get all of the y’s on one side of the equation and everything else on the other side. We get the y’s to the side of the equation where there are more y’s. For our equation, there are more y’s on the right, which means that we want to get all of the y’s to the right and all of the numbers to the left. This means that we need to move the 11 that is currently on the right side of the equation to the left. We are able to do this by taking 11 from both sides of the equation.
We want y to be the subject and not 4y. Therefore, we need to divide both sides of the equation by 4.
The final step is to replace y with f-1(x). Therefore, f-1(x) is:
*Note
It does not matter whether you swap your x and y variables at this stage or at the end.
If you swap them at the stage that I did, you would be making y the subject.
If you were to swap them later, you would make x the subject and when x is the subject, you would then swap x and y.
It does not matter when you swap the x and y values. I think that it is easier to swap them at the start because by swapping them at the start, it makes sure that you remember to swap them. Sometimes students forget to swap the x and y values at the end.
The function h(x) is given below:
Find the inverse function.
We are going to be finding the inverse function and the notation for the inverse function is h-1(x). The first step is to make y equal the function.
The next step is to swap x and y in the expression (you are able to swap these values later, but I think that it is easier to swap them now).
We now need to make y the subject of the expression. Currently the right side of the equation is a fraction with a y term on the denominator of the fraction. We want to get rid of the denominator, which we can do by multiplying both sides of the equation by the denominator; we multiply both sides of the equation by y – 2.
We then expand the brackets the left side of the equation.
The next step in making y the subject is to get all of the terms that contain y on one side of the equation and everything else on the other side. There is only one term that contains y, which is the term xy. This term is on the left and it is positive. Therefore, it makes sense to have the left side of the equation as the side for the terms that contain y and the right side as the side for everything else. The -2x that is currently on the left side of the equation does not contain y, which means that it needs to be moved to the right side of the equation. We move it to the right side of the equation by doing the opposite; we add 2x to both sides.
We want to find what y is equal to and not what xy is equal to. Therefore, we need to divide both sides of the equation by x.
The final step is to replace y with h-1(x). Therefore, h-1(x) is: