 Welcome back to our lecture series Math 1050, College Algebra for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misseldine. This is the first video in lecture eight in our series, which really is a continuation of lecture seven, which we saw previously, and it's continuation of our discussion about inverse functions. We saw in lecture seven that inverse functions swap the roles of x and y. That the x's become y-coordinates, the y-coordinates become x-coordinates when you switch from a function to its inverse. The input becomes output, the output becomes input when you switch to the inverse. Horizontal becomes vertical, vertical becomes horizontal when you switch to the inverse. The domain becomes the range, the range becomes the domain when you switch to the inverse, etc., etc. It turns out that this reversing of roles actually gives us an algorithm to find the inverse of some algebraically given function. Consider the function f of x equals 2x plus 3 here. Now, the term f of x is keeping track of the y-coordinates, so this really is the equation y equals 2x plus 3, and this is an equation for the function f. Well, to get the inverse function and to switch into the bizarro realm, we want to go to the inverse function, which is accomplished by swapping the x and y-coordinates. The y becomes an x, and then the x becomes a y. So if the equation y equals 2x plus 3 determines the function f, then x equals 2y plus 3 determines the function f inverse. And then we just proceed to solve, solve for y here, and that'll then give us the algebraic representation of this function. So we can accomplish this by minusing 3 from both sides. Notice we end up with 2y equals x minus 3, then divide both sides by 2, so the 2 cancels on the left-hand side, and then we end up with y equals x minus 3 over 2. If you want to distribute the fraction in the bottom, you can get 1 half x minus 3 halves, and there you get it. f inverse of x is equal to 1 half x minus 3 halves, which, when we played around with the function f of x equals 2x plus 3 in lecture 7, this was the function we found out to be its inverse. And this is how one actually finds the inverse, right? We checked the inverse previously, but how do you actually compute the inverse from scratch? You reverse the roles of x and y, and then you solve for y. That'll give you the formula each and every time. Knowing the solving for y can be a little bit tricky, depending on the algebraic nature of the function, but in some more videos in this lecture, we're going to practice some of the more complicated solving techniques as we compute inverse functions.