# Finding the inverse of function

• yojo95
In summary, the student is trying to solve for y in the equation y= f(x), but they get stuck because they need to solve for sinh-1 first.
yojo95

## Homework Statement

find the inverse, f(x) = 1 + e^sinh(x)

## The Attempt at a Solution

I am sorry but I never encounter this problem before and my teacher never showed us how to do these kind of problems, I have no idea what to do or how to start it out =[

then let $u = e^{x}$ and see if you can solve for u

so is it: (u - e^-x) / 2 = 0 ?
then: u = e^-x ?

yojo95 said:

## Homework Statement

find the inverse, f(x) = 1 + e^sinh(x)

## The Attempt at a Solution

I am sorry but I never encounter this problem before and my teacher never showed us how to do these kind of problems, I have no idea what to do or how to start it out =[
I seriously doubt that your teacher "never showed us how to do these kind of problems". Perhaps not with these particular functions- but the idea is the same as for linear functions. To find the inverse of any function f(x), write y= f(x), then "swap" x and y, x= f(y), and solve for y. If you have ever learned how to solve equations, you have learned how to do this.

(To shortcut any arguments, yes, some people learn to "first solve y= f(x) for x, then swap x and y. It's the same thing.)

$$y= f(x)= 1+ e^{sinh(x)}$$
becomes
$$x= 1+ e^{sinh(y)}$$

Solve for y by "backing out". Since 1 is added on the right, subtract 1 from each side:
$$x- 1= e^{sinh(y)}$$
Now we have an exponential on the right. The opposite of that is "ln" so take the natural logarithm of both sides:
[tex]ln(x-1)= sinh(y)[/itex]

What do you think we should do now?

since we want to get rid of sinh, do we take the inverse sinh, sinh-1, of both sides?

sinh-1(ln(x-1)) = sinh-1(sinh(y))
sinh-1(ln(x-1)) = y

## 1. What does it mean to find the inverse of a function?

Finding the inverse of a function means finding a new function that "undoes" the original function. In other words, if the original function takes an input and produces an output, the inverse function takes that output and produces the original input.

## 2. How do you find the inverse of a function?

To find the inverse of a function, you need to follow a set of steps. First, switch the x and y variables in the original function. Then, solve for y to get the new function. This new function is the inverse of the original function.

## 3. Why is it important to find the inverse of a function?

Finding the inverse of a function is important because it allows us to solve for the original input when given the output. It also helps us understand the relationship between the inputs and outputs of a function and can be useful in real-world applications.

## 4. Is every function invertible?

No, not every function has an inverse. For a function to be invertible, it must pass the horizontal line test, meaning that every horizontal line intersects the function at most once. If a function fails this test, it cannot have an inverse.

## 5. Can you find the inverse of a function algebraically and graphically?

Yes, you can find the inverse of a function both algebraically and graphically. Algebraically, you can follow the steps mentioned in the second question. Graphically, you can reflect the points of the original function over the line y = x to get the points of the inverse function.

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