Take the derivative of x+e^x and get 1+e^x > 0 for all x. So this function is strictly increasing and thus one-to-one. So y=x is the only set of solutions.
Skipped forward, saw that approach #2 was what I had in mind and wanted to check that I didn't miss anything. Sometimes we aren't that lucky with functions. 😂
This does not apply in the general case. y=x is a trivial set of solutions, but not always the only such solution set. Indeed, in the complex numbers, there are other less trivial solutions.
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Trivial in the reals, x = y, as x + e^x is a monotone, and thus invertible function.
Take the derivative of x+e^x and get 1+e^x > 0 for all x. So this function is strictly increasing and thus one-to-one.
So y=x is the only set of solutions.
This. Notice X and e^x are strictly increasing functions thus the function is bijective so an inverse exists and applying the inverse function x=y
Rearrange to get -1 = e^x - e^y / x-y which geometrically represents the slope of a secant line for the curve e^z. But it can never be negative.
f-1(x)= x-e^(x-w(e^x))
Yes, I think that solution works. I obtained f^-1(x) = ln(W(e^x)) from f(x) = x + e^x by using the substitution x = ln(g), then solving for g.
The solution can also take the form f^-1(x) = x - W(e^x).
Skipped forward, saw that approach #2 was what I had in mind and wanted to check that I didn't miss anything. Sometimes we aren't that lucky with functions. 😂
I know
It wasn't the main point of this video, but 0^0 is false. a^b * a^0 = a^b is true, but you can't divide both side by a^b, if a = 0.
I don't think that lim( x^x) _ as x--> 0 being 1 necessarily implies that 0^0 =1.
I was looking for a twist ending...
It should be pretty obvious that y=x.
It is less obvious that it might not be the general solution.
This does not apply in the general case. y=x is a trivial set of solutions, but not always the only such solution set.
Indeed, in the complex numbers, there are other less trivial solutions.
x=y ☝️🤓
F(x) = x + e^x is injective. So, the only solution is y=x.
Obvious solution y=x
y = x
Y can be 1 and e can be 2 and x can be 1
actually, e is a mathematical constant approximately equal to 2.718281828459045 and is equal to 1/0! + 1/1! + 1/2! + 1/3! + ...
f-¹(x) = ln(W[e^x])
X== y= any. No.
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9 minutes for this crap?