Question

If g(x)= 4+ x+ e^x, find g^-1 (5)

Answer 1

HINT: \(g(0) = 5\)

Answer 2

Even though this is totally beyond the scope of the question, you can algebraically invert this function.

Answer 3

Ok sorry sasha don't look cause this won't help you, sorry.

\[x=4+y+e^y\]\[x-4 = y+e^y\]\[e^{x-4} = e^ye^{e^y}\]\[W(e^{x-4}) = e^y\]\[\ln [W(e^{x-4})] = y\]
We could stop here but this is kind of ugly I think so let's rearrange this identity for Lambert W:
\[W(x)e^{W(x)} = x\]\[\ln[W(x)] +W(x)=\ln x\] So I can rewrite y now as:

\[y=x-4-W(e^{x-4})\]
I like this more, and we can go ahead and plug in x=5 now like they ask in the problem to get:

\[y(5)=5-4-W(e)=\]
So this is kinda cute and fun since this is the general solution to a problem most people think isn't solvable with algebra.

Answer 4

lol

@sasha.o do you get the first hint? if \(f(a)=b\) then \(f^{-1}(b)=a\)