How do i find the inverse?

Question

(3e^x-11)/(13e^x+11)

Answer 1

For

f(x) = (3e^x - 11) / (13e^x + 11)

To find the inverse, let y = f(x); that is, substitute f(x) for y.

y = (3e^x - 11) / (13e^x + 11)

Swap the x and y terms; you want to solve for y.

x = (3e^y - 11) / (13e^y + 11)

Multiply both sides of the equation by 13e^y + 11.

x(13e^y + 11) = 3e^y - 11

Expand the left hand side

13x e^y + 11x = 3e^y - 11

Now, move everything with e^y to the left hand side, while moving everything else to the right hand side.

13x e^y - 3e^y = -11x - 11

Factoring out e^y on the left hand side.

e^y (13x - 3) = -11x - 11

Divide both sides by (13x - 3), to isolate the e^y.

e^y = (-11x - 11) / (13x - 3)

Let's clean up that right hand side for a bit. I'm going to factor -1 out of the top and bottom.

e^y = [(-1) (11x + 11)] / [(-1) (3 - 13x)]

This cancels out the -1

e^y = (11x + 11) / (3 - 13x)

At this point we convert this to logarithmic form.

y = ln[(11x + 11) / (3 - 13x)]

Now, you make your concluding statement.

f^(-1)(x) = ln[(11x + 11) / (3 - 13x)]

Remember that what I assumed you were asking is the FUNCTIONAL inverse. The MULTIPLICATIVE inverse is something different and assumed by the previous answerers.

Answer 2

To find the inverse function, you can switch x and y and then solve for y.

Switch x and y,
x = (3e^y-11)/(13e^y+11)
x(13e^y+11) = 3e^y-11, times (13e^y+11) on both sides

Solve for e^y first,
e^y = 11(x+1)/(3-13x)

Solve for y,
y = ln(11)+ln[(x+1)/(3-13x)]

Answer 3

I assume you mean inverse function, not reciprocal.

Let y = (3e^x - 11)/(13e^x + 11)

To find the inverse function, switch the x and y variables and solve for x.

x = (3e^y - 11)/(13e^y + 11)
x(13e^y + 11) = 3e^y - 11
13xe^y + 11x = 3e^y - 11
13xe^y - 3e^y = - 11 - 11x
3e^y - 13xe^y = 11 + 11x
e^y(3 - 13x) = 11 + 11x
e^y = (11 + 11x)/(3 - 13x)

Taking the ln of both sides.

y = ln{(11 + 11x)/(3 - 13x)}
y = ln(11 + 11x) - ln(3 - 13x)

The inverse function then is:

f‾¹(x) = ln(11 + 11x) - ln(3 - 13x)

Answer 4

In order to find the inverse simply flip the fraction:

(13e^x+11)/(3e^x-11)

It is as simple as that.

Answer 5

Move what was in the numerator to the denominator and the denominator to the numerator.
(13e^x+11)/(3e^x-11)