This Article will show how to find the Inverse function of the Function of The Hyperbolic sine of x, by using the Definition that Sinh x = [e^x  e^(x)]/2.
From this method,the Inverse functions of the other five Hyperbolic functions can be found.
Things You'll Need
 Paper and
 Pencil.

In this article, we will be finding the inverse of f(x)=sinh x, which is equal to [e^xe^(x)]/2. The first step we will take is to set sinh x equal to y. Now, we will replace every 'x' with 'y', and every 'y' with 'x'. Please click on the image for a better understanding.

Now that we have x=[e^ye^(y)]/2, we must find what y is equal to, which is the inverse of the function. The first thing we will do is multiply both sides by 2, to remove the fraction. This will give us 2x=e^ye^(y). e^(x) can be rewritten as 1/(e^x), which will give us 2x=e^y(1/e^x). Now only one of the terms has a fraction. If we cross multiply, we can then multiply both sides by the denominator, eliminating fractions from the function. When we cross multiply the two e terms, we get: 2x=(e^2y1)/e^y. Please click on the image for a better understanding.

An equation that has fractions in it is more difficult to work with than one that doesn't. Because of that, we will try to remove any fractions in the function, before we do anything else. Why make things more difficult than they need to be? We have the function: 2x=(e^2y1)/e^y. To remove the fraction, we will multiply both sides of the equation by e^y. This will give us: 2xe^y=e^2y1. If we think of (e^y) as a variable, this can be in the form we need for the quadratic equation, which is 0=Ax²±Bx±C, once we subtract 2xe^y from both sides: 0=e^2y2xe^y1. The quadratic formula is x=[b±√(b²4ac)]/2a. Our A term is 1, our B term is 2x, our C term is 1, and our variable is e^y. When we put these terms in the quadratic equation, we get: e^y=[2x+√(4x²4(1)(1)]/2(1) = e^y=[2x+√(4x²+4)]/2. Please click on the image for a better understanding.

Before we can do anything else with the equation e^y=[2x+√(4x²+4)]/2, we need to simplify it. We will factor out the 4 in √(4x²+4), which will give us e^y=[2x+√4(x²+1)]/2. We will now separate it into two squareroots, instead of one: e^y=[2x+√4√(x²+1)]/2, which is equal to e^y=[2x+2√(x²+1)]/2. We can divide the 2 in the denominator by the 2's in both terms in the numerator, which will give us: e^y=x+√(x²+1). This function is much easier to work with. Please click on the image for a better understanding.

Finally, we will take the natural logarithm of both sides of the function, e^y=x+√(x²+1). We know that the natural logarithm of e raised to any power is that power. This will give us y=ln[x+√(x²+1]. This is the final answer for the Inverse Function of f(x)=sinh x. Please click on the image for a better understanding.