## anonymous one year ago If g(x)= 4+ x+ e^x, find g^-1 (5)

1. geerky42

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

2. Empty

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

3. Empty

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.

4. zzr0ck3r

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