Given f(x)=e^x - e^-x, find f^-1(x)?

Answer 1

Let y denote the inverse function.

Then, f(y)=x. We need to solve this equation for y.

First multiply

e^y-e^(-y)=x

by e^y. This gives us

e^(2y)-1=x e^y.

Because e^(2y)=(e^y)^2, the above equation is a quadratic in the variable e^y:

e^(2y)-x e^y-1=0.

Using the quadratic formula using the traditional a, b, c notation, we solve for e^(y): a=1, b=-x, c=-1

e^y=(x plus/minus sqrt(x^2+4))/2

Notice that x minus sqrt(x^2+4) is a negative number and the range of e^x is all positive numbers. Thus,

e^y=(x plus sqrt(x^2+4))/2

Now take the natural logarithm of both sides:

y=ln(e^y)=ln((x plus sqrt(x^2+4))/2)

Thus, the inverse of y=e^x-e^(-x) is
y=ln(e^y)=ln((x plus sqrt(x^2+4))/2).

As an exercise, you should use a graphing utility to graph each and confirm that their graphs are symmetric about the line y=x! (Setting your viewing window to [-10,10] by [-10,10] produces a nice result) :)

Answer 2

Notice that (e^x)(y) = e^2x - 1.

So (e^x)^2 - y(e^x) - 1 = 0.

Substitute u for e^x, and you have a quadratic:

u^2 - y u - 1 = 0.

The quadratic formula tells us that

u = (y ± sqrt(y^2 + 4))/2

Substitute e^x back in for u:

e^x = (y ± sqrt(y^2 + 4))/2

Then ln both sides:

x = ln((y ± sqrt(y^2 + 4))/2)

So for the inverse function, we have

f^-1(x) = ln((x ± sqrt(x^2 + 4))/2)

But... functions can't have a ± in them! A quick analysis, though, shows us that if we use the -, we will be taking the ln of a negative number, which we can't do (at least not in real numbers). So the sign we want is the +:

f^-1(x) = ln((x + sqrt(x^2 + 4))/2)

Answer 3

To get the inverse funtion you need to solve for x=f(y)
I don't believe there is a closed form solution to this function.

Answer 4

The answer is 42.

Answer 5

Your y = f(x) = 2*sinh(x) which is hyperbolic sine function. The inverse is just what it is: sinh^(-1)(2y). Inverse hyperbolic sine.

Answer 6

the inverse function is:
let y=e^^x-e^-x and then exchange x and y to get:
x=e^y-e^-y note that hyperbolic sine sinh(y)=(e^y-e^-y)/2
so x=2sinh(y) hence f^-1(x)=sinh^-1(x/2)

Answer 7

x=e^x/-e^x
x= -1