 In this equation, or in this problem here, we're going to be given a function f of x equals four x minus one over two x plus three, and we wanna find the range of domain of this function. All right, and we wanna do this purely algebraically. When it comes to finding the domain, remember the domain for a rational function, the things we should be looking out for are taking the square root of a negative and dividing by zero. There's no square roots here, but division by zero is a real possibility. What makes the denominator go to zero? Well, we consider the equation two x plus three equals zero. We'll minus three from both sides, get two x is equal to negative three, divide by two x equals negative three halves. And so this is then the forbidden value. The domain of our function f will be all real numbers x, such that x does not equal negative three halves. And so negative three halves feels like the kid who didn't get picked on the dodgeball team. Everyone else gets to play except for negative three halves. So sad, I'm sorry, but that's what it is. If you wanna write this in interval notation, we do negative infinity to negative three halves, union negative three halves, negative three halves to infinity. So every number except for negative three halves is part of the domain. So we know how to find the domain of a function algebraically, but how does one find the range? This one's a little bit trickier, right? Because the domain we're looking for, we're basically focused on what numbers can we not shove inside the machine? But what, now we have to ask, what number cannot come out of the machine? That's a little bit difficult, right? When you look at this thing, you have your machine, it's called f, we're analyzing what goes inside of it. We can see what fits inside of that hole. If we wanna know what comes out of it, then it makes sense to look at the inverse function because what comes out of the function is what goes inside of the inverse function. Look at the other hole. What cannot fit inside of that? And so if we knew what the inverse function was, we could actually find the domain of the inverse function very quickly. Remember that the domain of the inverse function, f inverse, this is the range of the function. So how about we find the domain of the inverse function? Well, that would require we compute the inverse function. Voila, we actually did that on a previous example. Here it is, yay. So we're not gonna do it again. We saw that the inverse function is negative one minus three x over two x minus four. So when will this function be undefined? It'll be undefined when this denominator goes to zero. So that means we have to solve the equation two x minus four equals zero. That makes the inverse function undefined. At four to both sides, you get two x equals four, divide by two, you get x equals two. That's your forbidden value. And so the domain of the inverse function is gonna be all numbers x such that x does not equal two. Or an interval notation, negative infinity to two, union two to infinity. So if we have a one-to-one function, finding its range just comes down to finding the domain of the inverse function. This is a nice thing because we can find domains of functions. So if we can find the inverse function, which we can now, and if we can find the domain, we can use that to find the range of functions. Now, that's a really great technique that I would recommend in general, but it does require the function be one-to-one, which good thing is linear fractional are always one-to-one functions. If your function's not one-to-one, we're gonna have to be a little bit more clever on finding the range. And that's a talk we're gonna have to have another day.