 Just like circular trigonometry, the six hyperbolic functions have inverses. We can talk about the inverse of hyperbolic sign, which we would call arc-sinch. We can talk about the inverse of hyperbolic cosine, which would be arc-cosh. We have hyperbolic arc tangent, hyperbolic arc cotangent, hyperbolic arc secant, and hyperbolic arc cosecant. We have inverses of each and every one of these. Now, you will call that cinch we defined earlier using the formula e to the x minus e to the negative x over two. So we defined hyperbolic sign using the natural exponential. It maybe comes as no surprise then that the inverse of hyperbolic sign can be described using logarithms. And I wanna describe to you exactly how one can see that. Let's carry this on below that if we have our function, if cinch inverse of x is equal to y by the inverse function property, this tells us that x is equal to cinch of y, which if we apply the definition, we're gonna get e to the y minus e to the negative y all over two. And then we can try to solve for x in the situation, excuse me, solve for y in the situation. Times both sides by two, we get two x equals e to the y minus e to the negative y. Now we're gonna move everything to one side of the equation. So we're just gonna subtract the two x from both sides. And we end up with the equation that e to the y minus two x minus e to the negative y is equal to zero. The next thing we're gonna do is we're gonna multiply both sides of this equation by e to the y. So we do it to the left-hand side and we're gonna do it to the right-hand side as well. This is the advantage of moving the right-hand side equal to zero. If you times by zero nothing happens. So distribute this e to the y through, we're gonna end up with an e to the two y or e to the y squared if you wanna think of it that way. You're gonna get a negative two x e to the y. And then e to the y times e to negative y is just gonna give you a one. So you get negative one that's equal to zero. So with this in mind, we can treat the left-hand side as a quadratic expression. It's a quadratic-like equation, which instead of the usual variable x, we're gonna have this e to the y right here. And so applying the quadratic equation, we get that e to the y is equal to, the coefficients here, you're gonna get that a equals one, b equals negative two x and c equals negative one for the quadratic formula. So you get a negative b, which will be two x, plus or minus the square root of b squared, which is a four x squared. You get a negative four ac. Well, a is just one and c is just negative one. So we're gonna get a plus four right there. And this all sits over a two a. So we just get two times one. So just a two right there. Trying to simplify this thing, notice we can factor out a four, leaving behind x squared plus one, for which then the square root of four is a two. So we get two x plus or minus two times the square root of x squared plus one all over two. You can factor the two out from the numerator and this will cancel out with the two in the denominator, thus simplifying to give us x plus or minus the square root of x squared plus one, which is equal to e to the y. Well, you can see here that if you take the negative sign, so there's sort of two possibilities that have to be considered here. You have e to the y is equal to x plus the square root of x squared plus one. And then there's the other situation we have e to the y is equal to x minus the square root of x squared minus one. Well, notice that if you take the square root of x squared, that's just gonna give you an x. So x minus x is gonna give you zero. An x differential function can't equal zero. Well, x squared plus one is a little bit bigger than x squared, which means the square root of x squared plus one is a little bit bigger than x. So this actually would be a negative number, which x-minusals can't be negative. So that possibly doesn't happen. Turns out this is the possibility we have to proceed with, for which then to solve for y, if you take the natural log of both sides, you'll end up with y equals the natural log of x plus the square root of x squared plus one, like so. Which coming back up to the top, that's exactly the formula we had for arc sense right here. Now I'm not gonna go through this calculation for the other five inverse hyperbolic functions, but we can see that the inverse of cosh is gonna look similar. It's gonna be the natural log of x plus the square root of x squared minus one. The hyperbolic arc tangent will be one half the natural log of one plus x over one minus x. Very similar is the formula of inverse hyperbolic cotangent, which is one half the natural log of x plus one over x minus one. The inverse of hyperbolic secant will be the natural log of one plus the square root of one minus x squared over x. And then the inverse of hyperbolic cosecant will be the natural log of one plus the square root of one plus x squared all over x. Now it's also important to mention that there are some domain restrictions in play here that arc sense will be defined for all real numbers, but the other functions will have restrictions to their domains. So arc cosh is only defined for values of x that are greater than or equal to one. Arc, hyperbolic arc tangent is only defined for x is between negative one and one. Arc cotangent hyperbolic is only defined for the absolute value of x is greater than one, which is just a fancy way of saying that x is less than negative one or x is greater than one. It turns out that hyperbolic arc secant is gonna be, x has to be between zero and one where one is included. And arc cosecant hyperbolic is defined for all real numbers except for zero. These is not something necessarily that's easy to memorize by all means. But if we explore the graphs of these functions for a second, we can understand exactly why we have these restrictions on their domains. Let's briefly discuss the graphs of the inverse hyperbolic functions. You can see the graph of cinch illustrated to you in orange on the screen right now. This graph is in fact a one-to-one function. As you look at any horizontal lines, it'll intersect the graph at exactly one location. So since the function is one-to-one, we can invert it and if you're reflected across the diagonal line, y equals x, which is about right here on the screen, you get a graph that looks like this, illustrated here in blue. You'll notice that these functions are in fact mirror images of each other or reflected across that diagonal line, y equals x. Well, some things that we can say, well, since the domain and range of cinch are all real numbers, we see that the domain and range of arc cinch are also all real numbers, although there is an inverse relationship going on here. As x approaches infinity, arc cinch will go towards infinity and as x approaches negative infinity, arc cinch will also approach negative infinity. What about the inverse of cosh? Well, that one's a little bit more tricky because cosh itself is not a one-to-one function. You can see that horizontal lines will intersect the graph in multiple locations like you see this right here. As such, in order to invert it, we have to restrict its domain. This is something we commonly do with the trig functions like sine and cosine. So if we turn that on for a second, notice that if we were just to take the right half of the cosh function, that will retain the entire range, one-to-infinity, but there's no more duplications. The duplications came from the left side of the graph, right? When we have things over here having the same wide coordinate as what's on the right-hand side. So we have to restrict the domain of cosh to be zero to infinity. And then in that situation, the graph is invertible in which case the inverse of cosh, that is arc cosh, will look like this blue curve right here. Cosh kind of looks like a square root or excuse me, it looks like kind of like an even monomial like a parabola. And so its inverse kind of looks like a square root, although instead of radical growth, this one will be much more like a logarithmic growth. It's very slow growing, but looks kind of like a square root, again with that logarithmic growth right there. Because of the restrictions we had a place on it, the domain of cosh inverse will be from zero, excuse me, from one-to-infinity, and the range will be from zero to infinity as well. We can see that as x approaches infinity, arc cosh will approach infinity very, very slowly, like logarithmically slow. And as you approach x equals one from the right, you'll be approaching y equals zero from above. Now consider the graph of hyperbolic tangent. You'll recall that this graph had some horizontal asymptotes at one and negative one. So while the domain is all real numbers for hyperbolic tangent, its range is from negative one to one, where one and negative one are not included in that. This is a one-to-one function, so we can invert it. And when we do so, we get this graph here in blue-ish-green, which is arc tangent, the hyperbolic arc tangent, which again, it looks like the reflection across the diagonal line y equals x. This function, it looks kind of like it's just an odd monomial, but when you zoom out, you see that there's some asymptotes going on here. These are in fact, vertical asymptotes. The vertical asymptotes will coincide with x equals one and x equals negative one, because when you take a function and switch to its inverse, horizontal asymptotes will turn into vertical asymptotes. So our function of hyperbolic arc tangent, its domain will be from negative one to one, where negative one and one are not included in the domain, but its range is gonna be all real numbers. Consider now our bump function that is hyperbolic secant. This function, much like Cosh, which we can see right there, right? These functions are reciprocals of each other. These functions, neither of these functions, Cosh or hyperbolic secant, these are not one-to-one functions. So if we want to have an inverse of hyperbolic secant, we have to restrict its domain. So considering that here, much like Cosh, there's a duplicity of y-coordinates if we get rid of the left-hand side of the graph, we actually get something that's now one-to-one and hence invertible. So we're gonna reflect that across the line y equals x and we get this blue function right here, which is the graph of hyperbolic arc secant, okay? Well, the hyperbolic secant had a horizontal asymptote at the x-axis. When you invert this, this shows us that hyperbolic arc secant will also get a asymptote. There'll be a vertical asymptote at the y-axis we see right here. And so because of this restriction, we see that the domain of hyperbolic arc secant is gonna be from zero to one, where one is included because we do have an x-intercept right here, zero is not included in the domain because of the vertical asymptote. But then the range of hyperbolic arc secant will be from zero all the way up to infinity. As x approaches zero from the right, hyperbolic arc secant will go towards infinity. And as we approach one from the left, hyperbolic arc secant will approach zero from above. You can now see on the screen illustrated in orange, the graph of hyperbolic cosecant. It's the reciprocal of cinch, right? This function does have a horizontal asymptote at the x-axis. It has a vertical asymptote at the y-axis as well. This function is one to one. I mean cinch was one to one and so was hyperbolic cosecant. As such, we can invert it and we get this blue curve, which looks very, very similar to hyperbolic cosecant. They're not the same one arc cosecant. It's a little bit different. They will have the same asymptotes though. Hyperbolic arc cosecant, it'll, as x approaches infinity, it'll approach zero from above. As x approaches negative infinity, it'll approach zero from below. As x approaches zero from the right, arc cosecant will go to a negative infinity, the hyperbolic one will. And as x approaches zero from the left, we see that hyperbolic arc cosecant will approach zero from the left. Last but not least is the hyperbolic cotangent, whose graph you can see on orange on the screen right now. This function is one to one. It's the reciprocal of hyperbolic tangent. These functions are one to one. So they are invertible. The inverse is gonna be hyperbolic arc cotangent, which you see there in the bluish green on the screen. We should point out that the hyperbolic cotangent function does have a vertical asymptote at x equals zero. As such, its inverse will have a horizontal asymptote at y equals zero, aka the x-axis. Likewise, the hyperbolic cotangent function, it has horizontal asymptotes at y one and y equals negative one. As a consequence, the arc cotangent function, the hyperbolic variant, will have vertical asymptotes at x equals one and x equals negative one. So now look at that here on the screen. So as x approaches infinity, arc cotangent hyperbolic will approach zero from above. As x approaches negative infinity, the function will approach zero from below. As x approaches one from the right, hyperbolic arc cotangent will approach infinity. As x approaches negative one from the left, this function will approach negative infinity. And then between one and negative one is domain's land. There's nothing that's outside the domain of our function here. So the domain of this function will be, it'll be x is less than negative one, strictly less than, or x is greater than negative one, which we can abbreviate that as the absolute value of x needs to be greater than one. That's the domain of this function. The range of this function will then be all real numbers except for zero because of the horizontal acetote at y equals zero, the x-axis.