 In this video we provide the solution to question number 13 for practice exam 1 for math 1050. We are given the 1 to 1 function f of x equals negative 3x minus 4 all over x minus 2. You don't have to prove it's 1 to 1, you can assume it. We want to compute the algebraic formula for the inverse function of f right here. Now remember the idea about the inverse function here, that you have your function f that's given in terms of its graph, the f of x is just the y-coordinate. This formula for f gives us y equals negative 3x minus 4 over x minus 2. Now as we switch to the inverse function, we switch the roles of x and y. Therefore the y variable becomes an x and the x variables become y's. So we get negative 3y minus 4 over y minus 2 like so. Now what our goal is, is we need to solve for the y-coordinate because that'll give us the formula for f inverse of x. We have two y's in play here, one's in the numerator, one's in the denominator. We need to get them together and so the first thing we're going to do is clear the denominators. They can't be combined together if one's in the denominator, one's in the numerator. So we times both sides by y minus 2 that cancels in on the left hand side. And then on the right hand side we get x times y minus 2. I'm going to go ahead and distribute that x there, so we get xy minus 2x. And then that's going to equal the right hand side, which is now 3y minus 4, like so. Our goal is to combine together the y's. We have a y right here and we have a y right here. So I'm going to move things from one side of the equation to the other. The negative 3y will move to the left hand side by adding 3y to both sides. The negative 2x will move to the right hand side by adding 2x to both sides. That gives us something like xy plus 3y is equal to 2x minus 4, like so. And now you'll notice because of how we organize things, there is a multiple of y on the left hand side. In fact, everything's a multiple of y. We have xy and we have 3y. We can factor the y out, giving us a y times x plus 3, and this is equal to 2x minus 4. And so we wanted to solve for y. That was our goal. So to do that, divide by the coefficient of y, which in this case is x plus 3, so that it cancels out. But what's good for the goose is good for the gander. We have to do the same thing that we did to the left hand side. We have to do that to the right hand side. So you get this x plus 3 over here. And so in the end, we end up with y is equal to 2x minus 4 over x plus 3. This gives us the formula for f inverse. But a common mistake that students make is they stop right here. If I just put a cute little box around here, I haven't yet answered the question. The question is, what is f inverse of x? You didn't tell me what f inverse of x is. So we need to make sure we are specific in our label. And we're saying that f inverse of x is this formula right here. If you just leave it as a y, we've kind of overloaded the meaning of y in this context. There's a y here. There's a y later on. You need to be very explicit. Don't leave anything up to the imagination here. When you're trying to prove that you know something, say it so that this is known. So you do need to write f inverse of x is, in this case, 2x minus 4 over x plus 3.