# Inverse Function: Finding x for y in f(x)

• Sweet_GirL
In summary, the conversation discusses finding the inverse function of f(x) = e^x - e^-x + 2 where x ≥ 0. The attempted solution involves replacing f(x) with y, but this is not a correct solution. It is suggested to use the fact that e^x + 2 + e^-x = (e^x/2 + e^-x/2)^2 and the definition of cosh(x) to find the inverse function.

#### Sweet_GirL

Inverse function (Edited)

## Homework Statement

Find the inverse function of :
$$f(x)=e^x-e^{-x}+2$$ where $$x \geq 0$$

## Homework Equations

All what I did is :
$$y=e^x-e^{-x}+e$$

## The Attempt at a Solution

How in Earth can I solve this for x ?

Last edited:
Sweet_GirL said:

## Homework Statement

Find the inverse function of :
$$f(x)=e^x+e^{-x}+2$$ where $$x \geq 2$$

## Homework Equations

All what I did is :
$$y=e^x+e^{-x}+e$$
This isn't right. All you have done is replace f(x) by y. The right side should have remained the same.

It's probably helpful to note that ex + 2 + e-x = (ex/2 + e-x/2)2, and also that cosh(x) = (1/2)(ex + e-x), where cosh(x) is the hyperbolic cosine of x.

Sweet_GirL said:

## The Attempt at a Solution

How in Earth can I solve this for x ?

## What is an inverse function?

An inverse function is a mathematical operation that "undoes" another function. It essentially reverses the input and output of a function. For example, if a function f(x) takes an input of x and gives an output of y, the inverse function would take an input of y and give an output of x.

## Why do we need to find x for y in inverse functions?

Finding x for y in inverse functions allows us to solve equations and problems that involve a dependent variable (y) and an independent variable (x). It also helps us understand the relationship between the input and output of a function.

## How do you find the inverse function of a given function?

To find the inverse function of a given function, you can use the steps: (1) replace f(x) with y, (2) interchange the positions of x and y, (3) solve for y, and (4) replace y with f-1(x). It is important to note that not all functions have an inverse function.

## What is the notation for inverse functions?

The notation for inverse functions is f-1(x), where "f" represents the original function and the "-1" indicates the inverse operation. It is important to note that this does not mean the inverse of f multiplied by -1.

## What are some real-life applications of inverse functions?

Inverse functions have various applications in fields such as physics, engineering, economics, and statistics. For example, they can be used to determine the speed of an object based on its position or to find the original value of a discounted price. They are also used in solving exponential and logarithmic equations.