Find Inverse of f(x) = x+2e(x)

[chrisparr]

Homework Statement

Let f(x)=x+2e(x). Find the inverse of f(x).

Homework Equations

e(a+b)=e(a)*e(b)
e(a-b)=e(a)/e(b)
ln(ab)=ln(a)+ln(b)
ln(a/b)=ln(a)-ln(b)
ln(a^b)=bln(a)

The Attempt at a Solution

Switch y and x so that
x=y+2e(y)
I tried applying ln to the functions at the start, subtracting y and then applying ln, applying ^2 to both sides but nothing seems to come of it. Any first suggestions?

 

[LCKurtz]

There is no "elementary algebra" way to solve that last equation for y.

 

[Architeuthis Dux]

First, let's rewrite the equation as y = x + 2e(x). This will make it easier to switch the x and y variables. Next, we can apply the natural logarithm to both sides of the equation:

ln(y) = ln(x + 2e(x))

We can use the property ln(a+b) = ln(a) + ln(b) to rewrite the right side of the equation as:

ln(y) = ln(x) + ln(2e(x))

Then, we can use the property ln(ab) = ln(a) + ln(b) to rewrite the right side again as:

ln(y) = ln(x) + ln(2) + ln(e(x))

Since ln(e(x)) = x, we can simplify further to:

ln(y) = ln(x) + ln(2) + x

Now, we can use the property ln(a^b) = bln(a) to rewrite the left side of the equation as:

ln(y) = ln(2x) + x

Finally, we can use the property e^(ln(x)) = x to rewrite the equation as:

y = e^(ln(2x) + x)

Therefore, the inverse of f(x) = x + 2e(x) is g(x) = e^(ln(2x) + x).