g(x) = ln sqrt (x+2)
Find the value of g^-1 (0)
How would I start? I'm stuck on this question for hours ! Please give me some hints. Thanks
Here's one way:
Let $\displaystyle y=g(x)=\ln\sqrt{x+2}$. To find the inverse, we swap x's and y's to get $\displaystyle x=\ln\sqrt{x+2}$. Now, we resolve the equation for y:
$\displaystyle x=\ln\sqrt{y+2}\implies e^x=\sqrt{y+2}\implies\dots\implies y=g^{-1}\left(x\right)=\dots$.
Then evaluate it at x=0.
Can you take it from here?
so to find the inverse you just switch the x and y and solve for y.
So if you have g(x) = y = ln sqrt (x+2)
you change it to x = ln sqrt (y+2) and solve for y:
x^e = sqrt(y+2)
x^2e = y+2
etc.
and if you want g inverse of 0, then you just plug in 0=x.
edit: woops, a minute too late. haha.
Or, since you are only asked for $\displaystyle g^{-1}(0)$, solve $\displaystyle 0= ln(\sqrt{x+2}}$ for x. Taking the exponential of both sides, $\displaystyle e^0= 1= \sqrt{x+2}$. Squaring both sides, 1= x+2 so x= -1. That shows that g(-1)= 0 so $\displaystyle g^{-1}(0)= -1$.