 Another important concept is the idea of an inverse function. So remember that a function has a unique output for any given input. Now, we can go further and say that a function is 1 to 1 if it has a unique input for any given output. For example, let's determine if f of x equals 3x plus 7 is a 1 to 1 function. Now, because we're going to be playing around with function values, it'll be convenient if we designate the function with a variable. So let's suppose y equals f of x. So the function will be 1 to 1 if any given output y comes from a unique input x. So let's try to find the input x that gives us an output y. Equals means replaceable. So if y equals f of x and f of x is 3x plus 7, then we can write down the equation. Since we want to find the input x, let's solve this equation for x. Then any given output y can be produced by the input x equal to y minus 7 over 3. So again, the question is whether or not any given output y comes from a unique value of x. Well, if I have a specific value of y, there's only one possible value for this expression, y minus 7 over 3. So this function is 1 to 1. How about this function? Again, suppose we let y equals f of x. The function will be 1 to 1 if any given output y comes from a unique value of x. So let's try to find the input x that gives us an output y. Equals means replaceable. So I'll write down the equation. Now, if y is a specific number, this is a quadratic equation which will in general have two solutions. And that means that most output values y will be produced by two input values x, and so this function is not 1 to 1. This leads to the idea of inverse function. If a function is 1 to 1, then, because it's a function, every input has a unique output, and because it's 1 to 1, every output comes from a unique input. And so if you know the output, you can find the input. And so we'll define the following. Let f be a function. The inverse function, written this way, is a function where f inverse of b is equal to a, if and only if f of a is equal to b. So there's a limited number of symbols available to us, so we have to reduce, reuse, recycle. And this notation for inverse function looks an awful lot like our notation for exponentials. It's important to recognize that this is not an exponent. Remember how you speak influences how you think, and so when you see this notation, you want to read this as f inverse of b. And so that leads to the following. There are three important things you want to remember about inverse function notation. First, f inverse does not mean 1 over f. The minus 1 indicates that it's an inverse and does not represent an exponent. Second, f inverse does not mean 1 over f. This minus 1 isn't an exponent. And most importantly, f inverse does not mean 1 over f. So let's find an inverse function. So definitions are the whole of mathematics, all else is commentary. If I want to find an inverse function, I can start out this way. Suppose f of a equals b, then f inverse of b should give you a. And that's straight out of the definition of what an inverse function is. Well, equals means replaceable. So if I know f of a is equal to b, I can replace f of a with equals means replaceable. Since I want to find f inverse of b, I really want to find a. So let's solve this equation for a. Equals means replaceable. So I have a, a is the same as f inverse of b, so I'll replace. And so now I have f inverse of b. Oh, wait a minute. We didn't want to find f inverse of b. We wanted to find f inverse of x. Well, that was a huge waste of effort. Or was it? If I want to find f inverse of x, why don't I replace b with x? And that gives us our inverse function, f inverse of x. How about a different function? Since this is a different function, we must do something completely different in order to find the inverse. Or not. We don't have to. We can rely on our definition of inverse function once again. Suppose f of a is equal to b, then f inverse of b is equal to a. This suggests I want to find f inverse, so I want to find a. So we'll solve this equation for a. Equals means replaceable. I have a, a is f inverse of b, so I'll replace. Now I actually want f inverse of x, so I'll replace. And there's my inverse function.