Find the formula for the inverse of the function
y =(e^x) / (1+8e^x)
to form the inverse , we interchange the x and y variables, so the inverse equation is
x = e^y/(1+8e^y)
now we have to solve that for y ....
e^y = x + 8x e^y
e^y - 8x e^y = x
e^y( 1 - 8x) = x
e^y = x/(1-8x)
y = ln(x/(1-8x) or lnx - ln(1-8x)
dada
To find the inverse of a function, we need to switch the roles of x and y in the equation and solve for y. So let's start by swapping x and y in the equation:
x = (e^y) / (1 + 8e^y)
Now, we will solve for y. To do this, we'll multiply both sides of the equation by (1 + 8e^y) to eliminate the fraction:
x(1 + 8e^y) = e^y
Next, distribute x to both terms on the left side:
x + 8xe^y = e^y
Now, let's isolate e^y by moving 8xe^y to the right side:
x = e^y - 8xe^y
Factor out e^y on the right side:
x = e^y(1 - 8x)
To solve for e^y, we'll divide both sides of the equation by (1 - 8x):
x / (1 - 8x) = e^y
Finally, we can rewrite e^y as ln(y):
ln(y) = x / (1 - 8x)
So the formula for the inverse of the function y = (e^x) / (1 + 8e^x) is:
ln(y) = x / (1 - 8x)