How do you find the inverse in f^-1(x) notation for f(x) = 3 radical x+3? I understand how to find the inverse of a function for most problems, but this one I just can't figure out. And for the 3 radical x+3, I don't know whether it's said 3 radical x+3 or 3 square root of x+3 so sorry for that if it's confusing.  Thanks

Expert Answers

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Find the inverse of `f(x)=3sqrt(x+3)` :

The typical way to find the inverse is to exchange x and y and solve for y:

`x=3sqrt(y+3)`

`x^2=9(y+3)`

`y+3=x^2/9`

`y=x^2/9-3`

But there is a problem. The original function had a domain restriction; `x>=-3` . We must take this into account. The range of the original function was `y>=0` so the domain of the inverse is `x>=0` .

If `f(x)=3sqrt(x+3)` then the inverse `f^(-1)(x)=1/9x^2-3,x>=0`

The graph of the function (green) and the inverse (red):

Note that as expected the graphs are reflections across the line y=x.

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