# Inverse Function f(x) = ln(e^2x + e^x + 1) Solution

• Master J
In summary, the conversation discusses finding the inverse of the function f(x) = ln(e^2x + e^x + 1). It is solved by converting the equation into a quadratic equation and taking the positive root, giving the inverse function f^-1(x) = ln((-1 + sqrt(4e^x - 3)) / 2).
Master J
f(x) = ln(e^2x + e^x + 1)

I want to find the inverse of this function.

I get:

e^y=e^2x + e^x + 1

If I say then that 1=e^0, then can I say y=2x + x? Therefore, x=y/3?

Master J said:
f(x) = ln(e^2x + e^x + 1)

I want to find the inverse of this function.

I get:

e^y=e^2x + e^x + 1

If I say then that 1=e^0, then can I say y=2x + x? Therefore, x=y/3?

You got it right as far as taking the exponential on both sides of the equation. From there it helps to notice that:

$$e^{2x}=\left(e^x\right)^2$$

Thus converting the entire right hand side into a quadric equation:

$$e^{y}=u^2+u+1$$

where $$u=\exp(x)$$ and whilst treating $$\exp(y)$$ as a constant, solve this with respect to u

Master J said:
f(x) = ln(e^2x + e^x + 1)

I want to find the inverse of this function.

I get:

e^y=e^2x + e^x + 1

If I say then that 1=e^0, then can I say y=2x + x? Therefore, x=y/3?
No, you can't. e2x+ ex+ e0 is NOT equal to e2x+ x+ 0

As Troels said, you can replace ex by u and get the equation ey= u2+ u+ 1 or u2+ u+ (1- ey)= 0 and solve that with the quadratic equation.

That gives
$$u= \frac{-1\pm\sqrt{4e^y- 3}}{2}$$
You might think that gives two solutions and so there is no "inverse" function (I did at first) but since u= ex, u cannot be negative: we must take the positive root:
$$u= e^x= \frac{-1+ \sqrt{4e^y- 3}}{2}$$
so
$$x= ln(\frac{-1+ \sqrt{4e^y- 3}}{2})$$

and the inverse function is
$$f^{-1}(x)= ln(\frac{-1+ \sqrt{4e^x- 3}}{2})$$

## 1. What is an inverse function?

An inverse function is a function that "undoes" the action of another function. In other words, if f(x) is a function, its inverse function is denoted as f^-1(x) and is defined as the function that, when applied to the output of f(x), gives back the input x. Inverse functions are useful in solving equations and understanding the relationship between two variables.

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

To find the inverse of a function, you must first switch the x and y variables in the equation. Then, solve for y. The resulting equation is the inverse function. In the case of f(x) = ln(e^2x + e^x + 1), the inverse function would be f^-1(x) = ln(e^2y + e^y + 1).

## 3. What is the purpose of finding the inverse of a function?

Finding the inverse of a function allows us to understand the relationship between two variables in a more meaningful way. It also allows us to solve equations involving the original function more easily. In some cases, the inverse function may have a more intuitive interpretation than the original function.

## 4. Can every function have an inverse?

No, not every function has an inverse. For a function to have an inverse, it must be a one-to-one function, meaning that each input has a unique output. In other words, no two inputs can have the same output. If a function is not one-to-one, it does not have an inverse. Additionally, some functions may have a restricted domain that prevents them from having an inverse.

## 5. How can you determine if a function has an inverse?

To determine if a function has an inverse, you can use the horizontal line test. If a horizontal line intersects the graph of the function at only one point, then the function is one-to-one and has an inverse. If the horizontal line intersects the graph at multiple points, then the function is not one-to-one and does not have an inverse.

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