 So another important idea in mathematics is known as an inverse function, and this emerges as follows. Sometimes I know the result of a function operation, but I want to know what I started with, what the initial value was. So as an example, suppose I know that the result of a number times 5 is 20, and I want to figure out what I started with, and my answer is going to be, well, let's defer that. But maybe I also know that the result of a number times 5 is 30, and I want to know what I started with. And likewise, a number times 5 is 1, and I want to know what I started with. And in every case, I could set up an equation and solve it. So I want to solve a number times 5 is 20, so I want to solve 5 times x equals 20, and I figure out what the solution is. And then I have this next problem. And I have a number times 5 is 30, so I want to solve 5 times x equals 30, and then I have a number times 5 equals 1, so I want to solve 5 times x equals 1. Because our functions are the same, they're all a number times 5, the equations that I get are also going to be the same. I'm looking for the solution of 5x equals 20, 5x equals 30, 5x equals 1. I'm going to be looking at very similar equations, and what this suggests is maybe I can solve a general equation and use it as a formula to solve all of our problems. So here, I may take a look at my general equation. A number times 5 is some constant value. Don't know what it is. And my answer is going to be the solution to 5 times x equals c. And I can solve this equation x equals c over 5, and now I have a formula that'll allow me to solve all these other equations as well. Whatever this is, over 5 is going to be my initial value. So I can fill those in 20 over 5, 30 over 5, 1 over 5, and I'm able to solve all three of these equations very quickly once I've solved this general equation. This is the basic idea behind any inverse function. So, given some function f of x, the inverse function written this way is a function with the property that if I apply the function to the function itself, I get back what I started, for all x that's in the domain of my original function. Now, one important caution here, this is the notation that we use for inverse function. You might look at this and say, oh, that's an exponent minus 1. This is not an exponent. This does not say that we're raising f to power negative 1. And you might say, well, that's kind of objectionable. Why do we use two things in the same way? And the simple answer to that is there's only so many letters and symbols in the English language. Why is it that this ti, well, ti, how do you pronounce ti? Well, it's ti, why is it that we pronounce it shah here? Why do we use this to represent the shah sound as well? And the answer is there's only so many symbols that we can work with. And likewise here, this negative 1 in the context of functions always, always, always, always, always refers to the inverse function. Well, here's a way that we can perceive what we can do is we can set up our generic equation f of x equals c. I can attempt to solve for x. I do want to verify that I have a function. I first and foremost, an inverse function is a function. So I do want to make sure that I have a function of c. So for example, let's take f of x equals 3x minus 7. So I'll set up a generic equation and I'll solve for x. So I want f of x equals c. So I have c equals my function value 3x minus 7. And I'll solve for x and how we solve for x. This kind of depends on the equation that we have. We'll do various things to find it. And here I have my expression x equals c plus 7 over 3. Now, to decide whether this is a function, again, the important definition of a function is that for any input value, I get at most one output value. So for any value of c, well, I'll note that c plus 7, well, there's only one value of c plus 7. And that means that when I divide c plus 7 by 3, there's only one value of c plus 7. There's only one value of c plus 7 divided by 3. So that tells me that any value of c will produce exactly one output value, c plus 7 over 3. And so this is a function. Now, what we call the input variable doesn't really matter. We could say that the inverse function f inverse of c is c plus 7 over 3. I could say that. It's traditional to use x as our input variable. So we're going to do that really complicated thing that we do. We're going to drop out our variable and leave a empty set of parentheses. So there it goes. I've dropped out the variable. I am left with an empty set of parentheses. And what I'd like to go in here is x. So again, whatever goes in one set of parentheses has to go in all the parentheses. So I'll drop an x in there, and there's my inverse function. All right. Well, let's take a look at another function. So f of x equals x squared plus 3x minus 7. We'll find the inverse function if it exists. So again, I'll set up a generic equation. f of x equals that. I'll let c equal that. And I'll try to solve for x. And a little analysis goes a long way. This is a quadratic equation. And so I want to get all the terms onto one side and then apply the quadratic formula. But again, a little analysis goes a long way. The plus or minus that we have in the quadratic formula guarantees that in almost every case we're going to have two solutions. And what this means is that when I solve this equation for x, I will generally have two different solutions. And what that means is that there's going to be no inverse function. That for any given value of c, I'll generally get two possible values of x. And that means I'm going to not have an inverse function. And an inverse function does not exist for f of x equals x squared plus 3x minus 7.