Inverse Function for (e^x)/(e^x + 1): Steps and Example Solutions

In summary, the problem asks to find the inverse of the equation y = (e^x)/(e^x + 1). The conversation involved switching the variables and solving for y, but difficulties arose when trying to simplify the equation. After some discussion and attempts, it was determined that the inverse could be found by multiplying/dividing by e^-x, resulting in the inverse x = -ln(1/x - 1).
  • #1
physstudent1
270
1

Homework Statement


Find the inverse:

y = (e^x)/(e^x + 1)

Homework Equations





The Attempt at a Solution



I switched x with y and solved for y but I ended up getting lne^y - lnx = lne^y +ln1 and then -lnx= ln1
 
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  • #2
could you show your work?

and I think you split up e^x+1 to lne^y+ln1? you can't do that.
 
  • #3
ohhh your right it should be

e^y = x(e^y+1)

then

lne^y = lnx + ln(e^y+1) but I'm still stuck from here
 
  • #4
hm..ok take the ln(e^y+1) to the right side and simplify by combining the ln's

edit: whops meant take lne^y to the right side, lnx to the left and you should be able to simplify it
 
  • #5
what is there to combine
 
  • #6
so get -lnx = ln(e^y + 1) - lne^y

then -lnx = ln((e^ y +1)/e^y) ? this doesn't seem like it helped now I am back to where i started.
 
  • #7
hm..no I multiplied by -1 on the RHS and LHS and got x=-lny but plugging that in I get y=1+y >.<
 
  • #8
could you show the steps I'm not seeing it
 
  • #9
wow did that wrong too >.>

my algebra is really bad right now for some reason...hm..try multiplying/dividing by [tex]\frac{e^{-x}}{e^{-x}}[/tex]

ok yes multiply/divide by that and you will find the inverse.
 
Last edited:
  • #10
so do that by the original equation before i start trying to find the inverse ?
 
  • #11
i got -ln(1/x -1) = y for the inverse would anyone agree ?
 
  • #12
yes and try to get x^-x all by itself on the RHS or LHS so you don't have something like ln(e^x+1)=y
 
  • #13
yep that's what I got, and you can check by plugging it in. also should be x=f(y).
 
  • #14
gotcha :)
 

1. What is an inverse function?

An inverse function is a function that "undoes" another function. In other words, if a function f(x) maps input values to output values, its inverse function f^-1(x) maps the output values back to the input values.

2. How do I find the inverse function for (e^x)/(e^x + 1)?

To find the inverse function for (e^x)/(e^x + 1), you can follow these steps:

  1. Write the original function as y = (e^x)/(e^x + 1).
  2. Switch the x and y variables: x = (e^y)/(e^y + 1).
  3. Solve for y by isolating it on one side of the equation. In this case, you can multiply both sides by (e^y + 1) to get x(e^y + 1) = e^y, then subtract x from both sides to get e^y + 1 = e^y - x.
  4. Solve for e^y, then take the natural logarithm of both sides to get ln(e^y + 1) = ln(e^y - x).
  5. Simplify using logarithm rules to get ln(e^y + 1) - ln(e^y) = ln(e^y) - ln(x).
  6. The left side simplifies to ln(1 + e^y) and the right side simplifies to ln(e^y/x).
  7. Now, solve for y by subtracting ln(e^y) from both sides and dividing by ln(x) to get y = ln(1 + e^y) - ln(x).
  8. Thus, the inverse function is f^-1(x) = ln(1 + e^x) - ln(x).

3. How do I check if my inverse function is correct?

You can check if your inverse function is correct by using the composition property. This means that when you plug the original function into the inverse function and vice versa, you should get back the original input. In this case, when you plug (e^x)/(e^x + 1) into f^-1(x) = ln(1 + e^x) - ln(x), you should get back x.

4. Can you provide an example solution for finding the inverse function?

Sure! Let's say we have the function f(x) = (e^x)/(e^x + 1). We can follow the steps outlined above to find the inverse function:

  1. Write the original function as y = (e^x)/(e^x + 1).
  2. Switch the x and y variables: x = (e^y)/(e^y + 1).
  3. Solve for y: e^y + 1 = e^y - x ➔ e^y = ln(1 + e^y) - ln(x).
  4. Thus, the inverse function is f^-1(x) = ln(1 + e^x) - ln(x).

So, the inverse function for f(x) = (e^x)/(e^x + 1) is f^-1(x) = ln(1 + e^x) - ln(x).

5. Are there any restrictions for finding the inverse function of (e^x)/(e^x + 1)?

Yes, there are some restrictions. In order for the inverse function to exist, the original function needs to be one-to-one, which means that each input has a unique output. In this case, the original function is one-to-one for all real numbers except for x = 0. Therefore, the inverse function exists for all real numbers except for x = 0.

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