 Consider the function f of x equals x minus four to the fourth power. You can see this graph illustrated in yellow right here. It kind of looks like a parabola, but someone just sculpted their parabola and ceramic flaps. They were about to put it in the oven and then, oh no, they dropped it on the floor and so it gets squished at the bottom of this quartic polynomial. It'll be a lot more flat than your typical parabola looks like. The x minus four suggests it's been shifted to the right by four units. So one, two, three, four, and we get the graph f of x equals x minus four to the fourth. So let's find the domain of this function for which the function that becomes invertible that it has an inverse, but we retain the original range, right? This is an issue we saw earlier when we talked about the function y equals x squared. y equals x squared is not one to one, so it doesn't have an inverse function. It's not invertible. We're not able to invert it. But if we shrink the domain, right? If we remove those parts of the graph that failed the horizontal line test, then we can make it a one to one function and therefore there will be an inverse function. So you'll notice with our function f right here, if we start drawing horizontal lines, much like the parabola, there's gonna be cases where a horizontal line hits it at two different places. So if we wanna make this function invertible, we have to get rid of the portion of the graph where there's repeats hitting horizontal lines. And just like the parabola, we're basically just gonna take off the left-hand side of the bowl right here. If we only take the right-hand side, this graph will be invertible. So essentially what I'm saying here is we're gonna restrict the domain of f to just be four to infinity. If we restrict it to that, then the function will be one to one. But what about the range here? The range of our function is gonna be zero to infinity, right? Look at the y-coordinates. You don't get anything below the x-axis, but you can get everything above the x-axis going off towards infinity, in fact. Now notice that if you take the left side of the graph and the right side together, the range is gonna be all non-negative numbers, zero to infinity. But if you only take the right side of the graph, you still get all the y-coordinate zero to infinity. So by throwing away the left side of the graph, you don't lose any of the y-coordinates. We're only throwing out duplicate y-coordinates after all. So we still retain the original range, but we've now restricted the domain so that it passes the horizontal line test. In this context, we are now able to invert the function. And so thus there is now an inverse function available for us. So when we look at the equation for f inverse, we're gonna take the equation for f, which is, so f was given by y equals x minus four to the fourth. And so when we switch to the inverse function, f inverse here, we're gonna get x equals y minus four to the fourth. Take the fourth root of both sides. We get the fourth root of x equals y minus four. At four to both sides, we see that f inverse of x is equal to the fourth root of x plus four. And which is, of course, the graph you see right here. It looks a lot like a square root function, mostly because the fourth root looks a lot like a parabola, right? If you reflect this curve across the diagonal line here, you're gonna get this curve right here. They're inverse functions of each other. And so now we can see that the domain of the domain of f inverse is going to be the range of f and then the range of f inverse is going to be the domain of f, the x and y switch roles when you do this. And so this is an important strategy to pay attention to, that if you have a function that you want to be one to one that's not, we can always restrict the domain down, retaining the original range, but we just throw out all the duplicates that show up inside of the domain. That is duplicates of y coordinates. Once you've shrunken the domain to something that's one to one, you can then invert it. And if you keep track of what did you throw out, then you're like, okay, if I come back and use the inverse function, like let's say you have an equation of the form, x squared equals four, right? When we work with x squared itself is not a one to one function, but if we restrict it only to positive numbers, it is one to one and we can take the square root of both sides, it's inverse function. We get x equals two. We solve the equation. But the thing is since the original function is not one to one, we have to then go back to the original domain here and remember, oh yeah, you have a positive value and you have a negative value. So when we solve the equation, we get plus or minus two. The reason that this equation has multiple solutions is because it's not a one to one function, it's a two to one function. And so if we can solve the equation when we restrict the one to one case, then you can use the duplicity principle to solve this. This is also a technique one uses in trigonometry because in trigonometry, your functions are actually infinity to one, right? They don't, they don't pass the horizontal line test, but you can still restrict the graph to a one to one interval. You solve it, this is where sine inverse comes from, sine inverse of x. This gives you sort of one solution to the equation, but then that solution represents all the other solutions when you consider how do you, how do you replicate that solution? What's the multiplicity? And like in trigonometry, you have to look for, you know, this right here would be the reference angle. I won't go into too much more about that as this series is not about trigonometry, but I want you to see this principle's widespread in mathematics. If you want to solve an equation, if the function involves or not one to one, make it be one to one, shove that square peg in the round hole, you do it, you solve the equation, but then after you solve it, you grieve for the fact that you shoved your hole, you know, you shoved the wrong peg into the hole. And so you have to pay a price for it not being perfect. And that might mean you have to pay attention to, okay, I have a reference solution of two, but that reference is both the positive two and negative two. And so we can use, we can use inverses even when the function's not one to one, so long as we restrict it to, we restrict it to a one to one function, even if only temporarily.