Calculus I - Inverse Functions
In this section we will define an inverse function and the notation used for inverse functions. means that there is a nice relationship between these two functions. f(−1)=3(−1)−2=−5⇒g(−5)=−53+23=−33=−1g(2)=23+23=43⇒f(43)=3( There is an interesting relationship between the graph of a function. Number 1 and 3 are correct but it's not that it gets rotated 90 degrees, the inverse is a reflection over the line y=X from the parent function. Now that we can find the inverse of a function, we will explore the graphs of . This relationship will be observed for all one-to-one functions, because it is a.
This is done to make the rest of the process easier. This is the step where mistakes are most often made so be careful with this step.
Intro to inverse functions (video) | Khan Academy
This work can sometimes be messy making it easy to make mistakes so again be careful. Most of the steps are not all that bad but as mentioned in the process there are a couple of steps that we really need to be careful with since it is easy to make mistakes in those steps. For all the functions that we are going to be looking at in this course if one is true then the other will also be true.
However, there are functions they are beyond the scope of this course however for which it is possible for only one of these to be true. This is brought up because in all the problems here we will be just checking one of them.
Intro to inverse functions
We just need to always remember that technically we should check both. However, it would be nice to actually start with this since we know what we should get.
This will work as a nice verification of the process. Here are the first few steps.
The next example can be a little messy so be careful with the work here. With this kind of problem it is very easy to make a mistake here.
That was a lot of work, but it all worked out in the end. We did all of our work correctly and we do in fact have the inverse. There is one final topic that we need to address quickly before we leave this section. Basically speaking, the process of finding an inverse is simply the swapping of the x and y coordinates. This newly formed inverse will be a relation, but may not necessarily be a function.
The inverse of a function may not always be a function!
The original function must be a one-to-one function to guarantee that its inverse will also be a function. A function is a one-to-one function if and only if each second element corresponds to one and only one first element.
Each x and y value is used only once. Use the horizontal line test to determine if a function is a one-to-one function. Remember that the vertical line test is used to show that a relation is a function. An inverse relation is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function.
If the graph of a function contains a point a, bthen the graph of the inverse relation of this function contains the point b, a.
Should the inverse relation of a function f x also be a function, this inverse function is denoted by f -1 x. If the original function is a one-to-one function, the inverse will be a function.
relationship between a function and its inverse - Mathematics Stack Exchange
If a function is composed with its inverse function, the result is the starting value. Think of it as the function and the inverse undoing one another when composed.
The answer is the starting value of 2.